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+Draft version February 3, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+A gap-sharing planet pair shaping the crescent in HD 163296: a disk sculpted by a resonant chain
+Juan Garrido-Deutelmoser
+,1, 2 Cristobal Petrovich
+,1, 3 Carolina Charalambous
+,4
+Viviana V. Guzm´an
+,1, 2 and Ke Zhang
+5
+1Instituto de Astrof´ısica, Pontificia Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, Chile
+2N´ucleo Milenio de Formaci´on Planetaria (NPF), Chile
+3Millennium Institute for Astrophysics, Chile
+4naXys, Department of Mathematics, University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium
+5Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter Street, Madison, WI 53706, USA
+ABSTRACT
+ALMA observations of the disk around HD 163296 have resolved a crescent-shape substructure at
+around 55 au, inside and off-center from a gap in the dust that extends from 38 au to 62 au. In this
+work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio
+≃ 4 : 3) of sub-Saturn-mass planets inside the gap, with the crescent corresponding to dust trapped
+at the L5 Lagrange point of the outer planet. This interpretation also reproduces well the gap in the
+gas recently measured from the CO observations, which is shallower than what is expected in a model
+where the gap is carved by a single planet. Building on previous works arguing for outer planets at
+≈ 86 and ≈ 137 au, we provide with a global model of the disk that best reproduces the data and
+show that all four planets may fall into a long resonant chain, with the outer three planets in a 1:2:4
+Laplace resonance. We show that this configuration is not only an expected outcome from disk-planet
+interaction in this system, but it can also help constraining the radial and angular position of the
+planet candidates using three-body resonances.
+Keywords: protoplanetary disks — planet–disk interactions — hydrodynamics — planets and satellites:
+dynamical evolution and stability — radiative transfer
+1. INTRODUCTION
+Substructures are ubiquitous in protoplanetary disks,
+particularly in the dust density distribution exhibited
+by high angular resolution observations (Andrews 2020;
+Bae et al. 2022). The Atacama Large Millimeter Ar-
+ray (ALMA) has revealed a variety of substructures,
+whereas a large population of rings and gaps are shown
+in continuum observations, to a lesser extent, in molec-
+ular line emissions (e.g., van der Marel et al. 2019).
+The advances in spatial resolution have been able to
+resolve non-axisymmetric substructures within gaps, in-
+cluding systems such as PDS 70 (Benisty et al. 2021),
+HD 163296 (Isella et al. 2018), HD 100546 (P´erez et al.
+2020), HD 97048 (Pinte et al. 2019), and LkCa 15 (Long
+et al. 2022). These substructures may be due to embed-
+ded planets induced by gravitational interactions (e.g.,
+Bae et al. 2022), which vary depending on the emis-
+sion shape.
+Point-like emissions are generally associ-
+ated with an accreting planet surrounded by a circum-
+planetary disk (CPD) (Perez et al. 2015; Szul´agyi et al.
+2018), while crescent shapes may be related to the sta-
+ble Lagrange points L4 and L5 of a star-planet system
+(Rodenkirch et al. 2021; Long et al. 2022) or vortices.
+These hypotheses frequently assume that a single sub-
+structure is caused by a single planet, often in a Jo-
+vian mass regime.
+However, we have recently shown
+in Garrido-Deutelmoser et al. (2022), that a pair of
+lower-mass and gap-sharing planets can sculpt compact
+and/or elongated vortices within the gap that last for
+several thousands orbits.
+The disk surrounding HD 163296 contains ringed
+structures in the mm-continuum (Isella et al. 2018) and
+several molecular tracers (Law et al. 2021; Zhang et al.
+2021). In particular, inside the dust density gap that ex-
+tends from 38 to 62 au, a crescent-shaped substructure
+resides at around 55 au (Teague et al. 2021). Recently,
+it was suggested that the emission comes from the dust
+trapped around the stable point L5 of a Jupiter mass
+planet orbiting at 48 au (Rodenkirch et al. 2021). Even
+though this method seems to reproduce the broad fea-
+tures of the dust continuum distribution, two aspects
+remain unclear:
+arXiv:2301.13260v1  [astro-ph.EP]  30 Jan 2023
+
+ID2
+Garrido-Deutelmoser, et al.
+1. the crescent feature resides at 55 au, off-center
+from the dust gap, while the L5 point is co-orbital
+to the planet at 48 au. Varying the planet’s eccen-
+tricity to account for this shift is unlikely to help
+as the stability of the crescent is damaged, leading
+to its prompt disruption.
+2. the Jupiter-mass planet needed to retain observ-
+able amounts of dust at L5 and open a wide enough
+gap in the dust, is expected to open a deep gap1
+(Duffell 2020). This prediction disagrees with the
+recent results provided by Zhang et al. (2021), who
+found that the dust density gap has a correspond-
+ing CO gap ∼ 10 times shallower than the predic-
+tions involving a Jupiter. The local gas depletion
+depends on the planetary mass (∝ m−2
+p ) (Kana-
+gawa et al. 2015), whereby opting for a lower mass
+planet to carve a shallower gap, may not produce
+a sufficient gravitational interaction to enforce the
+dust trapping at L5.
+In this work, we propose a that a compact pair of
+sub-Saturn-mass planets can solve these issues, simul-
+taneously accounting for the dust emission (the shifted
+crescent and dust rings) and shallow gap in the CO.
+This scenario is largely motivated by our recent work in
+Garrido-Deutelmoser et al. (2022) where we showed that
+a compact pair of gap-sharing planets generally lead to
+nonaxisymmetric substructures like that observed in HD
+163296.
+2. SETUP
+The hydrodynamics simulations and radiative trans-
+fer calculations in this work largely follow the scheme
+in Rodenkirch et al. (2021) and are only briefly sum-
+marized here. We carried out 2D hydrodynamic simu-
+lations using the FARGO3D multifluid code (Ben´ıtez-
+Llambay et al. 2019; Masset 2000; Weber et al. 2019)
+to produce gas and dust density distribution for a fidu-
+cial disk model. The resulting density maps are read
+into the RADMC3D code (Dullemond et al. 2012) to
+calculate the radiative transfer image at λ ∼ 1.25 mm.
+We use this image and the HD 163296 template (that
+contain all the technical properties of the observation)
+in the SIMIO2 package to achieve the synthetic ALMA
+observation comparable to the dust continuum observa-
+tion provided by Isella et al. (2018).
+1 An increase in the local viscosity to produce a much shallower gap
+comes at the expense of reducing the lifetime of the L5 crescent,
+or even prevent its formation in the first place (Rodenkirch et al.
+2021).
+2 https://www.nicolaskurtovic.com/simio
+2.1. Hydrodynamic Simulations
+The initial surface density profiles for the gas (sub-
+index g) and dust (sub-index d) are given by
+Σg/d = Σg/d,0
+� r
+r0
+�−0.8
+exp
+�
+−
+�
+r
+rc,g/d
+�γg/d�
+,
+(1)
+where we set r0 = 48 au, the initial surface density
+Σg,0 = 37.4 gr cm−2, the cut-off radius rc,g = 165 au,
+and the exponent γg = 1.
+Similarly, for the dust we
+set Σd,0 = Σg,0/100 = 0.374 gr cm−2, rc,d = 90 au,
+and γd = 2. We include the evolution for five indepen-
+dent dust species. These grains have sizes in cm of 0.02,
+0.071, 0.13, 0.26, and 1.92. We use an aspect ratio of
+h(r) = h0(r/r0)f with h0 = 0.05 and a flaring index
+f = 0.25, which implies a mid-plane temperature profile
+described by T = 25(r/r0)−0.5 K. This setup coincides
+with that from Rodenkirch et al. (2021).
+The disk extends from rin = 5 au to rout = 197 au,
+implying an initial disk mass of ∼ 0.15 M⊙. The compu-
+tational domain is composed of nr = 512 logarithmically
+spaced radial cells and nθ = 768 equally spaced cells in
+the azimuthal [0, 2π] domain. We include a radially vari-
+able viscosity of the standard parameter α (Shakura &
+Sunyaev 1973) as
+α(r) = αin − αin − αout
+2
+�
+1 + tanh
+�r − ξ
+σr0
+��
+,
+(2)
+where the inner and outer viscosities are αin = 1 × 10−4
+and αout = 5 × 10−3, ξ = 144 au indicate the midpoint
+of the transition and σ = 1.25 defines the slope. Similar
+to the values provided by Liu et al. (2018).
+A system of 4 planets was embedded. The location of
+the two outer ones is indicated in Teague et al. (2018)
+through kinematic detections. The third planet’s posi-
+tion (i.e., the inner planet of the outer pair) is strongly
+associated with the potential velocity kink reported by
+(Pinte et al. 2020). The two inner planets are tightly
+packed and their parameters (masses and orbits) were
+derived numerically guided by the results in Garrido-
+Deutelmoser et al. (2022), where it was found that the
+planets should be gap-sharing with forming vortices at
+their Lagrange points, implying a condition in the plan-
+etary separation3 of:
+∆a ≲ 4.6H ≃ 11.5 au,
+(3)
+where H the scale high of the disk. In turn, the masses
+are constrained by the width of the gap. After a few
+3 This expression has been tested for planets with masses near the
+thermal mass of Mth = M⋆h3 ∼ 0.25MJ ≃ 80M⊕.
+
+HD 163296: crescent and resonant chain
+3
+dozen simulations attempting to match disk morphol-
+ogy in continuum observations at the ∼ 48 au region,
+we choose to place the planets at a1 = 46, a2 = 54,
+a3 = 84.5, and a4 = 137 au with their respective masses
+of M1 = 85M⊕, M2 = 60M⊕, M3 = 0.4MJup, and
+M4 = 1MJup. The four bodies can gravitationally in-
+teract between them, but they do not feel the disk. We
+ran an extra model to compare against previous works,
+substituting a Jupiter at 48 au instead of the inner pack-
+age of planets. Both cases were evolved for 0.48 Myrs,
+equivalent to 2000 orbits of the innermost planet.
+2.2. Radiative Transfer
+We convert the 2D dust surface density into a 3D vol-
+ume density assuming the vertical approximation given
+by
+ρdj(r, φ, θ) = Σ(r, φ)
+√
+2πHdj
+exp
+�
+− z2
+2H2
+dj
+�
+,
+(4)
+where z = r cos(θ) and the dust settling follows the diffu-
+sion model Hdj =
+�
+Dz/(Dz + Stj)H, with Dz = 0.6α
+the vertical diffusion coefficient, and Stj the Stokes num-
+ber of the species j (Weber et al. 2022). We assume an
+intrinsic volume density for the particles ρs = 2 gr cm−3
+and a power law for the grain size distribution, such that
+n(a) ∝ a−3.5. We assumed a dust composition of 20%
+amorphous carbon, 20% water ices and 60% silicates,
+where the corresponding dust opacities were computed
+with the code provided by (Bohren & Huffman 1983).
+The polar direction is distributed in 64 equally spaced
+cells and extended in [80.6◦, 99.4◦] inclination domain.
+We use nphot = 108 photon packages to calculate the
+dust temperature, and nscatt = 107 photon packages to
+trace the thermal emission. We use a full anisotropic
+scattering with polarization treatment. The system is
+assumed to be at a distance of 101 pc with a central star
+of mass 1.9 M⊙ and effective temperature Teff = 9, 330
+K. The inclination is taken to be i = 46◦ and position
+angle PA= 133◦.
+2.3. Synthetic Observations
+We use SIMIO that contains a suit of functions for
+CASA 5.6.2. We select the template designed for HD
+163296 to create images with the same uv-coverage as
+observation from Isella et al. (2018). We set the rescale
+flux option in 0.4 to get similar intensities. In addition,
+we add simple thermal noise4 of level 12mJy to finally
+get RMS noise of 0.022 mJy beam−1.
+4 https://simio-continuum.readthedocs.io/en/main/tutorials/
+tutorial 3.html
+3. LOCAL GAP DEPLETION
+Figure 1 shows the surface density after ∼ 0.5 Myr
+(2,000 orbits) for the single-Jovian case (panel a) and the
+two sub-Saturns with masses 85M⊕ and 60M⊕ (panel
+b). The corresponding azimuthally-averaged profiles in
+panel (c) show that ∼ 95% of gas is depleted for the
+single-Jovian, while only ∼ 55% is depleted for the two-
+planet case. Despite of their lower masses the planets
+pair creates the same gap width as the Jovian.
+This
+shallower gaps for fixed gap width are expected in com-
+pact multi-planet systems due the planet lower masses
+(depth Σgap/Σ0 ∝ M −2
+p ) and angular flux transferred by
+the neighbouring planets (Duffell & Dong 2015; Garrido-
+Deutelmoser et al. 2022).
+As argued by Zhang et al. (2021), if a Jupiter-mass
+planet had opened the corresponding CO gap, it would
+be 10 times deeper than what is actually observed. In-
+stead, by embedding two planets we can alleviate these
+differences and largely reduce this discrepancy as shown
+in panel (c). Therefore, the depletion values would be
+closer to the results from observations. In order to bet-
+ter quantify this, we compare the CO column density
+gaps with the surface density from both models5. As
+shown in Figure 2, this approach reproduces the gas gap
+reasonably well in the two-planet case, largely matching
+the depths and widths with a small offset of the peaks
+by ∼ 4 au. In turn, the single-Jovian case is too deep
+compared to the observations as expected.
+Beyond the third planet at ∼ 85 au, neither of the two
+models (single Jovian or compact pair) is able to repro-
+duce the gas gaps.
+This was already noted in Zhang
+et al. (2021) when comparing with the models from
+Teague et al. (2021) and it may partly be explained by
+the presence of a CO snowline at 65 au. Outside the
+mid-plane CO snowline, CO freezes out at the disk mid-
+plane and therefore the CO gap properties (e.g., width
+and depth) may deviate from that of gas gaps due to
+vertical temperature and CO abundance variations. We
+recall that our work mostly focuses on the gap and cres-
+cent at ∼ 50 au where these issues can be more securely
+avoided.
+4. THE CRESCENT FEATURE
+A closely-packed planet pair can directly affect the
+gas around each other’s coorbital regions by deposit-
+ing angular momentum from wave steepening and sub-
+sequent shocks. These shocks may strengthen the vor-
+tencity around the stable L4 and L5 Lagrange points,
+5 The Σ profiles were processed under smooth function methodol-
+ogy described described in Appendix A.
+
+4
+Garrido-Deutelmoser, et al.
+100
+50
+0
+50
+100
+x [au]
+100
+50
+0
+50
+100
+y [au]
+(a)
+100
+50
+0
+50
+100
+x [au]
+(b)
+0
+30
+60
+90
+120
+150
+180
+r [au]
+0.1
+1.0
+� /
+0
+�
+(c)
+single-planet gap
+two-planets gap
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+log10( )
+[gr cm 2]
+Figure 1. Panels show a time evolution after 2000 orbits at 48 au (∼ 4.8 × 105 yrs). (a) Surface density Σ maps in log-scale for
+a single Jupiter planet at 48 au. (b) The same as (a), but for a two-planet system with 85M⊕ and 60M⊕ instead the Jupiter.
+The crosses denote the position of the planets and the white lines indicate their orbits. The disk rotates in a clockwise direction.
+(c) Azimuthally averaged Σ/Σ0 profiles for both models.
+thus enabling the effective gas and dust trapping for at
+least thousands of orbital periods (Garrido-Deutelmoser
+et al. 2022).
+The overdensity around either L4 or L5 for the inner
+and outer planet is a highly dynamic problem, with the
+most prominent structure chaotically alternating loca-
+tion. This said, we observe that for several combina-
+tions of parameters (surface density, aspect ratio, plan-
+etary masses, and so on), the outer planet often retains
+large amounts of material around L5. The depicted be-
+havior would leave an off-center substructure inside the
+gap that greatly reproduces the distinctive emission in
+the disk as shown in Figure 3. More quantitatively, our
+models matches the observed azimuthal extent of ∼ 45◦
+and the radial intensity profiles passing through the cres-
+cent (peaks and troughs inside the ring at ∼ 68 au; see
+Appendix B for more details).
+4.1. Dynamical behavior
+Figure 4 compares the gas and dust density distribu-
+tion for different dust grain sizes. All the dust species
+show a clear shared gap between the two inner planets.
+In the largest size, the co-orbital regions of each one
+display an overdensity at L5, but the more prominent
+substructure belongs to the second planet.
+This out-
+put has taken into account two conditions. The choice
+of planetary masses in the inner system must be lower
+for the body orbiting the substructure, and the pres-
+ence of the two outer planets. If we neglect either, the
+Lagrange points can still trap dust, but the mass dis-
+tribution may change to make some L4 or the inner L5
+the most prominent. As shown in Garrido-Deutelmoser
+et al. (2022), this evolution is highly dynamic and ir-
+regular so that the overdensities around L4 and L5 con-
+0
+30
+60
+90
+120
+150
+180
+210
+240
+r [au]
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Normalized gap depth
+CO gaps in Ke Zhang 2021
+single-planet gas gaps
+two-planets gas gaps
+Figure 2. Comparison of column density gap in C18O (2−1)
+line observation found by Zhang et al. (2021) with gas gaps
+derived from surface density profiles in this work.
+stantly change. However, the final morphology in our
+configuration comes from the early stages of evolution.
+Theoretically, the stable Lagrange point L5 is located
+at 60◦ from the planet at its trailing position.
+How-
+ever, our model shows that the center of the crescent is
+slightly shifted and can vary between 65◦ and 85◦ for
+reason we still do not understand and deserve further
+investigation. Accordingly, the position of the proposed
+planet will be also slightly shifted from the Lagrange
+point. Finally, we observe that azimuthal extent of the
+crescent remains roughly constant and equal to ∼ 45◦
+similar to the observations.
+4.2. Behavior for different dust species
+
+HD 163296: crescent and resonant chain
+5
+In our fiducial model, the vortensity for L5 of the outer
+planet is stronger than that of the inner one (not shown).
+Therefore, the dust accumulation is generally expected
+to be greater around the orbit of the outer planet for all
+sizes. This is especially true for small sizes that are well
+coupled to the gas and librate with larger amplitudes
+around L5 (panels b and c in Fig. 4). As the size of the
+grains increases (Stokes numbers approach unity, panels
+d to f), the dust distribution becomes compact toward
+the center of the Lagrange point (Montesinos et al. 2020)
+and we can even see some tenuous accumulation around
+the inner planet’s L5 point at 1.9 cm (panel f).
+5. A LAPLACE RESONANCE CHAIN
+Our fiducial simulation has 4 planets with period ra-
+tios P2/P1 = (54/46)3/2 = 1.27, (84.5/54)3/2 = 1.96,
+and (137/84.5)3/2 = 2.06.
+Therefore, the three outer
+planets lie near a 1 : 2 : 4 commensurability, which be-
+comes nearly exact (within 1%) if planet 3 changes from
+84.5 au to 86 au. This fact begs the question of whether
+disk-driven migration may have placed the planets in
+their current, near resonant, orbits6.
+From Figure 1, the planets carved relatively shallow
+gaps, so we may estimate the rate of orbital migration
+following Kanagawa & Szuszkiewicz 2020 as:
+τa ≡
+���a
+˙a
+��� ≃
+�M⋆
+Mp
+� �
+M⋆
+Σmina2p
+� h2
+p
+ΩK,p
+,
+≃ 0.4 Myr
+�10 gr cm−2
+Σmin
+� �100 au
+ap
+�1/2
+�2M⊙
+M⋆
+� �1MJ
+Mp
+� � hp
+0.1
+�2
+.
+(5)
+where Σmin corresponds to the local density at the base
+of the density gap. Using the fiducial planetary parame-
+ters and M⋆ = 1.9 M⊙, a fix aspect ratio h = 0.1 and the
+surface density constraints from Zhang et al. (2021, Ta-
+ble 5 therein), we observe that the migration timescales
+are all comparable to the age of the system making mi-
+gration a plausible scenario.
+As a proof of concept, in Figure 5 we show an N-body
+integration using REBOUND (Rein & Liu 2012) and pre-
+scribing the damping timescales τa and τe ≡ e/| ˙e| for
+each planet in order to mimic planet-disk interactions
+in the REBOUNDx library (Tamayo et al. 2019). We set
+τa/τe = 100 with τa computed using Eq. 5, see values
+for the orbital decay timescales in Table 1. We begin
+6 We note that a disk-driven migration may also lead to offsets
+from the exact commensurabilities by either wake-planet interac-
+tions (Baruteau & Papaloizou 2013), disk-driven precession (e.g.,
+Tamayo et al. 2015) or resonant repulsion (e.g., Papaloizou 2011).
+the simulations with the planets further away from their
+current positions and let them migrate due to their in-
+teraction with the gaseous component of the disk, where
+kinematic evidence has been detected in the gas at ∼ 260
+au (Pinte et al. 2020; Teague et al. 2021), and extensions
+in CO (2-1) up to ∼ 500 au (Zhang et al. 2021). The
+evolution shows that all planet pairs are captured into a
+long resonant chain after ∼ 0.5 × 106 yrs. These corre-
+spond to a two-body 4:3 mean-motion resonance (MMR)
+for the innermost planets, and a double 2:1 - 2:1 MMR
+for the two outer pairs, finally leading to the libration
+of the following three-body angles:
+φ123 = 3λ1 − 5λ2 + 2λ3, and
+(6)
+φ234 = 2λ2 − 6λ3 + 4λ4.
+(7)
+Both φ123 and φ234 have small-amplitude libration
+(∼ 2◦) and their libration centers are 187◦ and 218◦, re-
+spectively. Note that when the four planets reach the re-
+ported semi-major axis at approximately the same time
+∼ 106 yrs (gray vertical line in panel b), they are al-
+ready captured in the two- and three-planet resonances,
+showing that the proposed configuration with our hydro-
+dynamical simulations presented in the previous sections
+is possible.
+5.1. Predicting the position angle (PA) of the planet
+candidates using 3-body resonances
+Because the orbits of planets i are coplanar and nearly
+circular (ei ≲ 0.07), the mean longitudes λi are close to
+the true longitudes ϖi +fi and will likely librate around
+the same angles. Defining an arbitrary reference frame,
+rotated by PA0 we write the position angles (PAs) as
+PAi = ϖi + fi + PA0, and define the following three-
+body angle, frame-independent7, combinations:
+PA123 = 3PA1 − 5PA2 + 2PA3,
+and
+(8)
+PA234 = 2PA2 − 6PA3 + 4PA4.
+(9)
+From panel (e) in Figure 5, we observe that these an-
+gles librate around PA123 ∼ 185◦ and PA234 ∼ 210◦ sim-
+ilar to φ123 and φ234, but with larger amplitudes (near
+100◦ in both cases). This is expected due to the non-zero
+eccentricities.
+Assuming that we know two angles, say PA2 due to
+the crescent and PA3 due to a velocity kink, we can use
+the above relations to constrain PA1 and PA4 as:
+PA1 ∼ 1
+3PA123 + 5
+3PA2 − 2
+3PA3,
+and
+(10)
+PA4 ∼ 1
+4PA234 − 1
+2PA2 + 3
+2PA3.
+(11)
+7 Since the critical angles satisfy the D’Alembert property, these
+combinations are independent of the reference frame.
+
+6
+Garrido-Deutelmoser, et al.
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+RA [arcsec]
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+Dec [arcsec]
+180◦
+90◦
+0◦
+270◦
+ALMA  Band 6
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+RA [arcsec]
+Synthetic Observation
+0.01
+0.1
+0.4
+1.0
+2.0
+Intensity [mJy beam 1]
+Figure 3. Band 6 (λ ∼1.25 mm) comparison between dust continuum image from ALMA observation (Isella et al. 2018) and
+our synthetic observation after ≈ 4.8 × 105 yrs. The synthesized beam are the same for both images (0.038′′ × 0.048′′, 82.5◦),
+represented by white ellipse at the bottom left corner for each image. The synthetic image is projected with an inclination
+i = 46◦ and a position angle PA= 133◦. The enumerated white dots indicate the position of potential planets associated with:
+(1) and (4) resonance angles in Laplace chains (see §5.1), (2) Lagrange point L5 from the simulation, (3) velocity kink reported
+by Pinte et al. (2020). The white dashed lines denote the orbits of planets in the simulation. The right bottom Cartesian
+coordinate describe the prescription to estimate the azimuthal angles. The disk rotates in a clockwise direction.
+Our model indicates that planet 2 (54 au) is ∼ 75◦ ±
+10◦ ahead of the crescent center8, which corresponds to
+PA2 ≃ 32◦.
+In addition, as reported by Pinte et al.
+(2020), the planet 3 at 86 au has PA3 ≃ 357◦ associated
+with a velocity kink. Thus, our model predicts that the
+planets at 46 au and 137 au should have position angles
+of PA1 ∼ 237◦ and PA4 ∼ 212◦ respectively (see the left
+panel in Figure 3).
+Quite recently9, Alarc´on et al. (2022) localized strong
+kinematic deviation in C I line emission. The position
+of this structure lies inside the gap at 48 au, which
+azimuthally coincides with our predicted planet 1 at
+PA1 ∼ 237◦. We note that this predicted planet differs
+from the one proposed by Isella et al. (2018) and in-
+correctly quoted by Alarc´on et al. (2022) as coincident
+with the outflow. The reason is that the disk rotates
+in a clockwise direction so the proposed planet invoked
+to explain the crescent as a L5 feature, similar to Ro-
+denkirch et al. (2021), will actually show ahead of the
+crescent. In this way, the C I deviation cannot be ex-
+plained by a co-rotational planet that is also responsible
+8 Due to the clockwise rotation of the disk, our angle convention
+PA is given the coordinate axes at the bottom of Figure 3.
+9 After the submission of our manuscript to the journal.
+Table 1. Migration rate estimates
+—
+—
+—
+—
+ap
+Mp
+Σmin [gr/cm/cm]
+τa [Myr]
+46 au
+85 M⊕
+12
+1.8
+54 au
+60 M⊕
+19
+1.5
+84.5 au
+127 M⊕
+9.3
+1.1
+137 au
+317 M⊕
+4.2
+0.8
+Note—The values of Σmin are taken from Table 5 in
+Zhang et al. (2021), except for the innermost one
+provided by our model.
+for the crescent, unless the dust accumulation around
+L4 becomes more prominent than that of L5, which is
+unlikely (Rodenkirch et al. 2021; Garrido-Deutelmoser
+et al. 2022).
+5.2. Resonances in other systems
+We remark that resonances may be a common out-
+come in these young systems, including the embedded
+planets in PDS 70 (Bae et al. 2019), as well as young,
+but disk-free systems, like HR 8799 also in a long reso-
+
+2
+3
+1
+4HD 163296: crescent and resonant chain
+7
+100
+50
+0
+50
+100
+y [au]
+Gas
+(a)
+0.2 mm
+(b)
+100
+50
+0
+50
+100
+y [au]
+0.7 mm
+(c)
+1.3 mm
+(d)
+100
+50
+0
+50
+100
+x [au]
+100
+50
+0
+50
+100
+y [au]
+2.6 mm
+(e)
+100
+50
+0
+50
+100
+x [au]
+1.9 cm
+(f)
+0
+50
+100
+150
+0.0
+0.5
+1.0
+1.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+2
+3
+0
+1
+2
+3
+4
+5
+0
+10
+20
+30
+Figure 4. Face-on gas and dust surface density Σ from the
+hydrodynamic model after ≈ 4.8×105 yrs (2000 orbits at 48
+au). The panels correspond to different fluids. The crosses
+denote the position of the planets and the white dashed lines
+indicate their orbits. The disk rotates in a clockwise direc-
+tion.
+nance chain involving four planets (Go´zdziewski & Mi-
+gaszewski 2020).
+Similar to our work, a compact multi-planet system
+has been proposed using the axisymmetric dust gaps
+and rings of HL Tau (ALMA Partnership et al. 2015),
+where a resonant configuration may promote the sys-
+tem’s dynamical stability (Tamayo et al. 2015). In our
+case, we use not only the system’s migration history and
+dust rings and gaps, but also add the constraints from
+the crescent shape structure and the CO gas emission.
+6. CONCLUSIONS
+We have provided a global model for HD 163296 with
+four planets (semi-major axes in the range of 40 − 140
+au) that can reproduce the rings and gaps in the dust
+continuum and the shallow gaps in the gas constrained
+by the CO emission. A key ingredient in our model is
+the presence of two sub-Saturn-mass planets near the
+4:3 resonance opening the gap at ∼ 48 au, where the
+Time [Myr]
+1.3
+1.6
+1.9
+2.2
+2.5
+period ratio
+2 : 1
+4 : 3
+(a)
+P4/P3
+P3/P2
+P2/P1
+Time [Myr]
+0
+100
+200
+300
+400
+a [au]
+(b)
+a4
+a3
+a2
+a1
+Time [Myr]
+0.00
+0.03
+0.06
+0.09
+e
+(c)
+e4
+e3
+e2
+e1
+Time [Myr]
+0
+90
+180
+270
+360
+3pl [deg]
+187
+218
+(d)
+123
+234
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Time [Myr]
+0
+90
+180
+270
+360
+PA3pl [deg]
+185
+210
+(e)
+PA123
+PA234
+Figure 5.
+Potential migratory history of the four-planet
+system locking the planets in a long orbital resonance chain
+leaving the outer three planets near a consecutive 2:1 com-
+mensurability and the innermost pair near 4:3 (panels a and
+b). The eccentricities remain small after the capture (panel
+c) and the three-body resonant angles φ123 = 3λ1−5λ2+2λ3
+and φ234 = 2λ2 − 6λ3 + 4λ4 undergo small-amplitude libra-
+tion (panel d). The bottom panel exhibits the corresponding
+combinations of the position angles in the system: PA123 =
+3PA1 − 5PA2 + 2PA3 and PA234 = 2PA2 − 6PA3 + 4PA4.
+crescent corresponds to the L5 Lagrange point of the
+outer planet at 54 au.
+We show that the four-planet system may be part of
+a long resonance chain with the inner two in a 4:3 MMR
+and the outer three in a 1:2:4 Laplace resonance chain,
+consistent with a history of convergent migration within
+the disk. Our proposed three-body resonances allow to
+relate the planetary radial and angular positions, and
+based on the crescent location at 55 au and the proposed
+location by Pinte et al. (2020) for the planet at ≃ 86 au,
+our model predicts two planets: i) a sub-Saturn at 46
+au and PA ∼ 237◦; ii) a Jovian at 137 au and PA ∼
+212◦(Figure 3).
+Overall, our work shows that tightly-spaced planetary
+systems, often found at small orbital distances in tran-
+
+XXXXXX8
+Garrido-Deutelmoser, et al.
+siting surveys, may leave detectable imprints in proto-
+planetary disks at much larger separations.
+Acknowledgements The authors would like to thank
+Andrew Youdin, Kaitlin Kratter, Diego Mu˜noz, Matt
+Russo, Pablo Ben´ıtez-Llambay, Sim´on Cassasus, and Xi-
+mena S. Ramos for helpful discussions that improved the
+quality of this work and Juan Veliz for his support with
+the cluster logistics. Finally we thank the anonymous re-
+viewer for the thorough and useful report. J.G. acknowl-
+edge support by ANID, – Millennium Science Initiative
+Program – NCN19 171 and FONDECYT Regular grant
+1210425. The Geryon cluster at the Centro de Astro-
+Ingenieria UC was extensively used for the calculations
+performed in this paper. BASAL CATA PFB-06, the
+Anillo ACT-86, FONDEQUIP AIC-57, and QUIMAL
+130008 provided funding for several improvements to
+the Geryon cluster.
+C.P. acknowledges support from
+ANID Millennium Science Initiative-ICN12 009, CATA-
+Basal AFB-170002, ANID BASAL project FB210003,
+FONDECYT Regular grant 1210425, CASSACA grant
+CCJRF2105, and ANID+REC Convocatoria Nacional
+subvencion a la instalacion en la Academia convocatoria
+2020 PAI77200076. C.C. acknowledges FNRS Grant No.
+F.4523.20 (DYNAMITE MIS-project). V.V.G. acknowl-
+edges support from FONDECYT Regular 1221352,
+ANID project Basal AFB-170002, and ANID, – Mil-
+lennium Science Initiative Program – NCN19 171. K.Z.
+acknowledges the support of the Office of the Vice Chan-
+cellor for Research and Graduate Education at the Uni-
+versity of Wisconsin – Madison with funding from the
+Wisconsin Alumni Research Foundation.
+Software: Fargo3D (Ben´ıtez-Llambay et al. 2019),
+Numpy (van der Walt et al. 2011), Matplotlib
+(Hunter 2007). Rebound (Rein & Liu 2012), Re-
+boundX (Tamayo et al. 2019), RADMC3D (Dullemond
+et al. 2012).
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+10
+Garrido-Deutelmoser, et al.
+APPENDIX
+A. GAS GAP CALCULATION
+r [au]
+101
+102
+[gr cm 2]
+(a)
+6
+10
+20
+30
+60
+100
+200
+300
+0
+30
+60
+90
+120
+150
+180
+r [au]
+101
+102
+[gr cm 2]
+(b)
+6
+10
+20
+30
+60
+100
+200
+300
+�
+convolved
+�
+regions {r}
+smooth fit
+Figure 6. Black line denote the convolved surface density
+profile of models and grey dashed line the respective smooth
+function. Panel (a) and (b) represent the single-planet case
+and two-planet case respectively. The crosses indicate the
+position of the planets.
+In Zhang et al. (2021) a smooth function was sub-
+tracted from NCO column density profiles to better char-
+acterize substructures in the residual values. To com-
+pare these results with our models, we follow the same
+procedure. First, the Σ maps from hydrodynamic simu-
+lations were convolved with a circular Gaussian beam of
+0.15′′, which has the same size as MAPS CO (2-1) line
+observations. Then, their azimuthally averaged profiles
+were interpolated every 2 au. In addition, the radial re-
+gion {r [au] : 0 < r0 < 35, 59 < r1 < 72, 98 < r2 <
+110, r3 > 170} was selected to describe the gaps. Both
+were taken as input for the smoothfit10 module. The
+Figure 6 shows the outputs of smoothed profile repre-
+10 https://pypi.org/project/smoothfit/
+30
+60
+90
+120
+150
+180
+r [au]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+�Intensity
+�
+[mJy beam 1]
+180◦
+90◦
+0◦
+270◦
+350◦
+300◦
+ALMA Band 6
+Synthetic Obs.
+Figure 7. Azimuthally averaged intensity profiles for syn-
+thetic and ALMA observations after ≈ 4.8 × 105 yrs. The
+crosses denote the semi-major axes of the planets. The insert
+shows the ALMA observation in Band 6 with contours that
+reproduce the crescent and rings as well as the angular slice
+used for the azimuthal average denoted by ˆφ.
+sented by the grey dashed line and the convolved sur-
+face density profile in black lines. The cyan dots denote
+the regions in which the function acts. The Figure 2
+shows the residual between lines to provide a reasonable
+comparison with CO gaps observations.
+B. RADIAL INTENSITY
+We quantify the intensity around the substructure re-
+gion of our synthetic model with the ALMA observation.
+First, we deproject the images obtaining a face-on view
+to convert them to polar coordinates and then generate a
+radial profile by taking the azimuthal average between
+PA of 300◦ and 350◦. This extension fully covers the
+emission from the crescent. The results are shown in
+the Figure 7, which is accompanied by a diagram show-
+ing the angular slice.
+Figure 7 show that radial intensity through the cres-
+cent region reaches amplitudes higher than those ob-
+served by a factor of 1.2 at 55 au. The emission from
+the substructure is clearly off-centered on the gap and
+resolved in spatial resolution, showing a gap in intensity
+between it and the ring. The first ring reproduces the
+intensities in a good way, while the second is noticeably
+1.7 times fainter.
+
+HD 163296: crescent and resonant chain
+11
+2 /3
+/3
+0
+/3
+2 /3
+ [rad]
+0.02 cm
+Feel Disk = YES
+0.02 cm
+Feel Disk = NO
+30
+40
+50
+60
+70
+r [au]
+2 /3
+/3
+0
+/3
+2 /3
+ [rad]
+1.9 cm
+30
+40
+50
+60
+70
+r [au]
+1.9 cm
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+ [gr cm
+2]
+0.02
+0.04
+0.06
+0.08
+0.10
+ [gr cm
+2]
+Figure 8. Dust surface density Σ for 0.02 cm and 1.9 cm
+grain sizes. The crosses denote the position of the planets.
+The upper and bottom white rectangles, indicate the La-
+grange points L4 and L5 respect to the outer planet.
+C. EFFECT FROM DISK GRAVITY ACTING ON
+PLANETS
+We briefly test whether turning on the full disk-planet
+interaction may lead to morphological changes in the
+structure of the crescent. We recall that in our fiducial
+simulation (see §2.1), while the disk do feel the planets’
+gravity, the planets do not feel the disk.
+We perform two-planet simulations considering only
+the inner planet pair near the 4:3 commensurability (46
+au and 55 au) for up to 2000 orbits of the inner planet.
+The initial density has been reduced by a factor of 100
+to avoid significant migration. In figure 8 we show the
+density distribution for two dust fluids of 0.02 cm and
+1.9 cm grain sizes in two cases: the full disk-planet in-
+teraction is considered (left panels, displaying a slight
+inward migration at the ∼ 10% level), and the disk grav-
+ity acting on the planets is ignored (right panels, with
+no migration). Despite of the slight orbital migration,
+we do not observe any significant changes regarding the
+amount and distribution of captured material at the L4
+and L5 Lagrange points.
+

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+Published as a conference paper at ICLR 2023
+GENEFACE:
+GENERALIZED
+AND
+HIGH-FIDELITY
+AUDIO-DRIVEN 3D TALKING FACE SYNTHESIS
+Zhenhui Ye1∗, Ziyue Jiang1∗, Yi Ren2, Jinglin Liu1, JinZheng He1, Zhou Zhao1†
+1Zhejiang University
+{zhenhuiye,jiangziyue,jinglinliu,jinzhenghe,zhaozhou}@zju.edu.cn
+2Bytedance
+[email protected]
+ABSTRACT
+Generating photo-realistic video portrait with arbitrary speech audio is a crucial
+problem in film-making and virtual reality. Recently, several works explore the us-
+age of neural radiance field in this task to improve 3D realness and image fidelity.
+However, the generalizability of previous NeRF-based methods to out-of-domain
+audio is limited by the small scale of training data. In this work, we propose Gene-
+Face, a generalized and high-fidelity NeRF-based talking face generation method,
+which can generate natural results corresponding to various out-of-domain audio.
+Specifically, we learn a variaitional motion generator on a large lip-reading cor-
+pus, and introduce a domain adaptative post-net to calibrate the result. Moreover,
+we learn a NeRF-based renderer conditioned on the predicted facial motion. A
+head-aware torso-NeRF is proposed to eliminate the head-torso separation prob-
+lem. Extensive experiments show that our method achieves more generalized and
+high-fidelity talking face generation compared to previous methods1 .
+1
+INTRODUCTION
+Audio-driven face video synthesis is an important and challenging problem with several applications
+such as digital humans, virtual reality (VR), and online meetings. Over the past few years, the com-
+munity has exploited Generative Adversarial Networks (GAN) as the neural renderer and promoted
+the frontier from only predicting the lip movement Prajwal et al. (2020)Chen et al. (2019) to gener-
+ating the whole face Zhou et al. (2021)Lu et al. (2021). However, GAN-based renderers suffer from
+several limitations such as unstable training, mode collapse, difficulty in modelling delicate details
+Suwajanakorn et al. (2017)Thies et al. (2020), and fixed static head pose Pham et al. (2017)Taylor
+et al. (2017)Cudeiro et al. (2019). Recently, Neural Radiance Field (NeRF) Mildenhall et al. (2020)
+has been explored in talking face generation. Compared with GAN-based rendering techniques,
+NeRF renderers could preserve more details and provide better 3D naturalness since it models a
+continuous 3D scene in the hidden space.
+Recent NeRF-based works Guo et al. (2021)Liu et al. (2022)Yao et al. (2022) manage to learn an
+end-to-end audio-driven talking face system with only a few-minutes-long video. However, the
+current end-to-end framework is faced with two challenges. 1) The first challenge is the weak
+generalizability due to the small scale of training data, which only consists of about thousands-many
+audio-image pairs. This deficiency of training data makes the trained model not robust to out-of-
+domain (OOD) audio in many applications (such as cross-lingual Guo et al. (2021)Liu et al. (2022) or
+singing voice). 2) The second challenge is the so-called ”mean face” problem. Note that the audio
+to its corresponding facial motion is a one-to-many mapping, which means the same audio input
+may have several correct motion patterns. Learning such a mapping with a regression-based model
+leads to over-smoothing and blurry results Ren et al. (2021); specifically, for some complicated
+∗Authors contribute equally to this work.
+†Corresponding author
+1Video samples and source code are available at https://geneface.github.io
+1
+arXiv:2301.13430v1  [cs.CV]  31 Jan 2023
+
+Published as a conference paper at ICLR 2023
+audio with several potential outputs, it tends to generate an image with a half-opened and blurry
+mouth, which leads to unsatisfying image quality and bad lip-synchronization. To summarize, the
+current NeRF-based methods are challenged with the weak generalizability problem due to the lack
+of audio-to-motion training data and the ”mean face” results due to the one-to-many mapping.
+In this work, we develop a talking face generation system called GeneFace to address these two
+challenges. To handle the weak generalizability problem, we devise an audio-to-motion model to
+predict the 3D facial landmark given the input audio. We utilize hundreds of hours of audio-motion
+pairs from a large-scale lip reading datasetAfouras et al. (2018) to learn a robust mapping. As for
+the ”mean face” problem, instead of using the regression-based model, we adopt a variational auto-
+encoder (VAE) with a flow-based prior as the architecture of the audio-to-motion model, which
+helps generate accurate and expressive facial motions. However, due to the domain shift between
+the generated landmarks (in the multi-speaker domain) and the training set of NeRF (in the target
+person domain), we found that the NeRF-based renderer fails to generate high-fidelity frames given
+the predicted landmarks. Therefore, a domain adaptation process is proposed to rig the predicted
+landmarks into the target person’s distribution. To summarize, our system consists of three stages:
+1 Audio-to-motion. We present a variational motion generator to generate accurate and expressive
+facial landmark given the input audio.
+2 Motion domain adaptation. To overcome the domain shift, we propose a semi-supervised ad-
+versarial training pipeline to train a domain adaptative post-net, which refines the predicted 3D
+landmark from the multi-speaker domain into the target person domain.
+3 Motion-to-image. We design a NeRF-based renderer to render high-fidelity frames conditioned
+on the predicted 3D landmark.
+The main contributions of this paper are summarized as follows:
+• We present a three-stage framework that enables the NeRF-based talking face system to enjoy
+the large-scale lip-reading corpus and achieve high generalizability to various OOD audio. We
+propose an adversarial domain adaptation pipeline to bridge the domain gap between the large
+corpus and the target person video.
+• We are the first work that analyzes the ”mean face” problem induced by the one-to-many audio-
+to-motion mapping in the talking face generation task. To handle this problem, we design a varia-
+tional motion generator to generate accurate facial landmarks with rich details and expressiveness.
+• Experiments show that our GeneFace outperforms other state-of-the-art GAN-based and NeRF-
+based baselines from the perspective of objective and subjective metrics.
+2
+RELATED WORK
+Our approach is a 3D talking face system that utilizes a generative model to predict the 3DMM-
+based motion representation given the driving audio and employs a neural radiance field to render
+the corresponding images of a human head. It is related to recent approaches to audio-driven talking
+head generation methods and scene representation networks for the human portrait.
+Audio-driven Talking Head Generation
+Generating talking faces in line with input audio has
+long attracted the attention of the computer vision community. Earlier works focus on synthesizing
+the lip motions on a static facial imageJamaludin et al. (2019)Tony Ezzat & Poggio (2002)Vou-
+gioukas et al. (2020)Wiles et al. (2018). Then the frontier is promoted to synthesize the full headYu
+et al. (2020)Zhou et al. (2019)Zhou et al. (2020). However, free pose control is not feasible in these
+methods due to the lack of 3D modeling. With the development of 3D face reconstruction tech-
+niquesDeng et al. (2019), many works explore extracting 3D Morphable Model (3DMM)Paysan
+et al. (2009) from the monocular video to represent the facial movementTero Karras & Lehtinen
+(2017)Yi et al. (2020) in the talking face system, which is named as model-based methods. With
+3DMM, a coarse 3D face mesh M can be represented as an affine model of facial expression and
+identity code:
+M = M + Bidi + Bexpe,
+(1)
+2
+
+Published as a conference paper at ICLR 2023
+Driving 
+Audio
+HuBERT
+Features
+Variational Motion Generator
+𝑧~𝑁 0,1
+Enhanced
+Latent
+Generated
+Landmark
+Domain Adaptative Post-net
+Conv 1D
+BN
+ReLU
+x N
++
+Refined
+Landmark
+3DMM NeRF Renderer
+Head-NeRF
+Torso-NeRF
+Rendered
+Head
+Rendered
+Frame
+WaveNet-
+like Decoder
+Flow-based 
+Prior
+Figure 1: The inference process of GeneFace. BN denotes batch normalization.
+where M is the average face shape; Bid and Bexp are the PCA bases of identity and expression; i
+and e are known as identity and expression codes.
+By modeling the 3D geometry with 3DMM, the model-based works manage to manipulate the head
+pose and facial movement. However, 3DMM could only define a coarse 3D mesh of the human
+head, and delicate details (such as hair, wrinkle, teeth, etc.) are ignored. It raises challenges for
+GAN-based methods to obtain realistic results. Recent advances in neural rendering have created
+a prospect: instead of refining the geometry of 3DMM or adding more personalized attributes as
+auxiliary conditions for GAN-based renderers, we could leave these delicate details to be modeled
+implicitly by the hidden space of the neural radiance field.
+Neural Radiance Field for Rendering Face
+The recent proposed neural radiance field
+(NeRF)Mildenhall et al. (2020)Kellnhofer et al. (2021)Pumarola et al. (2021)Sitzmann et al. (2019)
+has attracted much attention in the human portrait rendering field since it could render high-fidelity
+images with rich details such as hair and wrinkles. For instance, Sitzmann et al. (2019) presents
+a compositional NeRF for generating each part of the upper body. NerFaceGafni et al. (2021) and
+Pumarola et al. (2021) propose pose-expression-conditioned dynamic NeRFs for modeling the dy-
+namics of a human face. EG3DChan et al. (2022) proposes a hybrid explicit–implicit tri-plane
+representation to achieve fast and geometry-aware human face rendering. HeadNeRFHong et al.
+(2022) proposes a real-time NeRF-based parametric head model.
+Several works have also applied NeRF in the audio-driven talking face generation task. Zhang et al.
+(2021) devise an implicit pose code to modularize audio-visual representations. AD-NeRF Guo
+et al. (2021) first presents an end-to-end audio-driven NeRF to generate face images conditioned on
+Deepspeech Hannun et al. (2014) audio features. Recently, SSP-NeRF Liu et al. (2022) proposed
+a semantic-aware dynamic ray sampling module to improve the sample efficiency and design a
+torso deformation module to stabilize the large-scale non-rigid torso motions. DFA-NeRF Yao et al.
+(2022) introduces two disentangled representations (eye and mouth) to provide improved conditions
+for NeRF. To achieve few-shot training, DFRFShen et al. (2022) conditions the face radiance field
+on 2D reference images to learn the face prior, thus greatly reducing the required data scale (tens
+of seconds of video) and improve the convergence speed (about 40k iterations). However, all of the
+previous NeRF-based work focuses on better image quality or reducing the training cost, while the
+generalizability to out-of-domain audio is relatively an oversight.
+Our GeneFace could be regarded as bridging the advantages of the aforementioned two types of
+works. Compared with previous 3DMM-based methods, our work could enjoy good 3D naturalness
+and high image quality brought by the NeRF-based renderer. Compared with previous end-to-end
+NeRF-based methods, we improve the generalizabity to out-of-domain audio via introducing a gen-
+erative audio-to-motion model trained on a large lip reading corpus.
+3
+GENEFACE
+In this section, we introduce our proposed GeneFace. As shown in Fig. 1, GeneFace is composed of
+three parts: 1) a variational motion generator that transforms HuBERT features Hsu et al. (2021) into
+3D facial landmarks; 2) a post-net to refine the generated motion into the target person domain; 3)
+a NeRF-based renderer to synthesize high-fidelity frames. We describe the designs and the training
+process of these three parts in detail in the following subsections.
+3
+
+Published as a conference paper at ICLR 2023
+Large Video Corpus
+Deep 3D Recon
+3DMM Landmark
+HuBERT
+Features
+WaveNet-like 
+Encoder
+WaveNet-like 
+Decoder
+Flow-based 
+Prior
+𝜇, 𝜎
+Variational Motion Generator
+Generated 
+Landmark
+Pretrained
+SyncNet
+Training of Audio2motion
+Audio
+Figure 2: The structure of variational motion generator. Dashed arrows means the process is only
+performed during training; and only the dashed rectangle part is used during inference.
+3.1
+VARIATIONAL MOTION GENERATOR
+To achieve expressive and diverse 3D head motion generation, we introduce a variational auto-
+encoder (VAE) to perform a generative and expressive audio-to-motion transform, namely the vari-
+ational motion generator, as shown in Fig. 2.
+Audio and motion representation
+To better extract the semantic information, we utilize Hu-
+BERT, a state-of-the-art ASR model, to obtain audio features from the input wave and use it as the
+condition of the variational motion generator. As for the motion representation, to represent detailed
+facial movement in Euclidean space, we select 68 key points from the reconstructed 3D head mesh
+and use their position as the action representations. Specifically,
+LM3D = {(M − M)i|i ∈ I},
+(2)
+where LM3D ∈ R68×3, M and M are the 3DMM mesh and mean mesh defined in Equation (1), I
+is the index of the key landmark in the mesh. In this paper, we name this action representation 3D
+landmarks for abbreviation.
+Dilated convolutional encoder and decoder
+Inspired by WaveNet, to better extract features from
+the audio sequence and construct long-term temporal relationships in the output sample, we design
+the encoder and decoder as fully convolutional networks where the convolutional layers have incre-
+mentally increased dilation factors that allow its receptive field to grow exponentially with depth.
+In contrast to previous works, which typically divide the input audio sequence into sliding windows
+to obtain a smooth result, we manage to synthesize the whole sequence of arbitrary length within
+a single forward. To further improve the temporal stability of the predicted landmark sequence, a
+Gaussian filter is performed to eliminate tiny fluctuations in the result.
+Flow-based Prior
+We also notice that the gaussian prior of vanilla VAE limits the performance of
+our 3D landmark sequence generation process from two prospectives: 1) the datapoint of each time
+index is independent of each other, which induces noise to the sequence generation task where there
+is a solid temporal correlation among frames. 2) optimizing VAE prior push the posterior distribution
+towards the mean, limiting diversity and hurting the generative power. To this end, following Ren
+et al. (2021), we utilize a normalizing flow to provide a complex and time-related distribution as the
+prior distribution of the VAE. Please refer to Appendix A.1 for more details.
+Training Process
+Due to the introduction of prior flow, the closed-form ELBO is not feasible,
+hence we use the Monte-Carlo ELBO loss Ren et al. (2021) to train the VAE model. Besides,
+inspired by Prajwal et al. (2020), we independently train a sync-expert Dsync that measures the
+possibility that the input audio and 3D landmarks are in-sync, whose training process can be found in
+Appendix A.2 . The trained sync-expert is then utilized to guide the training of VAE. To summarize,
+the training loss of our variational motion generator (VG) is as follows:
+LVG(φ, θ, ϵ) = −Eqφ(z|l,a)[log pθ(l|z, a)]+KL(qφ(z|l, a)|pϵ(z|a))−Eˆl∼pθ(l|z,c)[log Dsync(ˆl)] (3)
+where φ, θ, ϵ denote the model parameters of the encoder, decoder and the prior, respectively. c
+denotes the condition features of VAE. The ground truth and predicted 3D landmarks are represented
+by l and ˆl, respectively.
+4
+
+::.DPublished as a conference paper at ICLR 2023
+HuBERT
+Features
+Large Video Corpus
+Variational
+Motion
+Generator
+HuBERT
+Features
+Target Person Video
+Domain Adaptive 
+Post-net 
+Generated Landmark
++
++
+Refined Landmark
+MLP-based
+Disc. 
+as neg. 
+sample
+as neg. 
+sample
+GT target person 
+3DMM Landmark
+MSE
+as positive sample
+Training of PostNet
+Pretrained
+SyncNet
+Figure 3: The training process of Domain Adaptative Post-net.
+3.2
+DOMAIN ADAPTIVE POST-NET
+As we train the variational motion generator on a large multi-speaker dataset, the model can general-
+ize well with various audio inputs. However, as the scale of the target person video is relatively tiny
+(about 4-5 minutes) compared with the multi-speaker lip reading dataset (about hundreds of hours),
+there exists a domain shift between the predicted 3D landmarks and the target person domain. As a
+consequence, the NeRF-based renderer cannot generalize well with the predicted landmark, which
+results in blurry or unrealistic rendered images. To this end, A naive solution is to fine-tune the
+variational generator in the target person dataset. The challenge is that we generally only have a
+short personalized video, and the generalizability of the model may be lost after the fine-tuning.
+Under such circumstances, we design a semi-supervised adversarial training pipeline to perform a
+domain adaptation. To be specific, we learn a post-net to refine the VAE-predicted 3D landmarks into
+the personalized domain. We consider two requirements for this mapping: 1) it should preserve the
+temporal consistency and lip-synchronization of the input sequence; 2) it should correctly map each
+frame into the target person’s domain. To fulfill the first point, we utilize 1D CNN as the structure
+of post-net and adopt the sync-expert to supervise the lip-synchronization; for the second point, we
+jointly train an MLP-structured frame-level discriminator that measures the identity similarity of
+each landmark frame to the target person. The detailed structure of the post-net and discriminator
+can be found in Appendix A.3.
+Training Process
+The training process of post-net is shown in Fig.3. During training, the MLP
+discriminator tries to distinguish between the ground truth landmark l′ extracted from the target
+person’s video and the refined samples Gω(ˆl) generated from the large-scale dataset. We use the
+LSGAN loss to update the discriminator :
+LD(δ) = Eˆl∼pθ(l|z,c)[(Dδ(PNω(ˆl)) − 0)2] + El′∼p′(l)[(Dδ(l′) − 1)2]
+(4)
+where ω and δ are the parameters of the post-net PN and discriminator D. l′ is the ground truth
+3DMM landmark of the target person dataset, and ˆl is the 3D landmarks refined by the post-net.
+As for the training of post-net, the post-net competes with the discriminator while being guided by
+the pre-trained sync-expert to maintain lip synchronization. Besides, we utilize the target person
+dataset to provide a weak supervised signal to help the adversarial training. Specifically, we extract
+the audio c′ of the target person video for VAE to predict the landmarks ˆl′ ∼ pθ(l|z, c′) and en-
+courage the refined landmarks PNω(ˆl′) to approximate the ground truth expression l′. Finally, the
+training loss of post-net is:
+LPN(ω) = Eˆe∼pθ(l|z,c)[(Dδ(PNω(ˆl)) − 1)2] + Eˆl∼pθ(l|z,c)[Dsync(ˆl)]
++Eˆl′∼pθ(l|z,c′)[((PNω(ˆl′) − l′)2]
+(5)
+3.3
+NERF-BASED RENDERER
+We obtain a robust and diverse audio-to-motion mapping through the variational motion generator
+and post-net. Next, we design a NeRF-based renderer to render high-fidelity frames conditioned on
+the predicted 3D landmarks.
+5
+
+DPublished as a conference paper at ICLR 2023
+Training of NeRF
+Target Person Video
+Deep 3D Recon
+3DMM Landmark
+Landmark
+Encoder
+Head Color
+Encoder
+Head
+Pose
+Head
+NeRF
+Torso
+NeRF
+Rendered
+Head
+Rendered
+Frame
+3DMM NeRF Renderer 
+Figure 4: The training process of NeRF-based renderer.
+3D landmark-conditioned NeRF
+Inspired by Guo et al. (2021), we present a conditional NeRF
+to represent the dynamic talking head. Apart from viewing direction d and 3D location x, the
+3D landmarks l will act as the condition to manipulate the color and geometry of the implicitly
+represented head. Specifically, the implicit function F can be formulated as follows:
+Fθ : (x, d, l) → (c, σ)
+(6)
+where c and σ denote the color and density in the radiance field. To improve the continuity between
+adjacent frames, we use the 3D landmarks from the three neighboring frames to represent the facial
+shape, i.e., l ∈ R3×204. We notice that some facial landmarks only change in a small range, which
+numerically raises challenges for NeRF to learn the high-frequency image details. Therefore, we
+normalize the input 3D landmarks point-wisely, which is necessary to achieve better visual quality.
+Following the setting of volume rendering, to render each pixel, we emit a camera ray r(t) = o+t·d
+in the radiance field, with camera center o, viewing direction d. The final color C is calculated by
+aggregating the color c along the ray:
+C(r, l; θ) =
+� tf
+tn
+σθ(r(t), l) · cθ(r(t), l, d) · T(t)dt
+(7)
+where tn and tr is the near bound and far bound of ray r; cθ and σθ are the output of the implicit
+function Fθ, T(t) is the accumulated transmittance along the ray from tn to t, which is defined as:
+T(t) = exp(−
+� t
+tn
+σθ(r(τ))dτ)
+(8)
+Head-aware Torso-NeRF
+To better model the head and torso movement, we train two NeRFs to
+render the head and torso parts, respectively. As shown in Fig. 4, we first train a head-NeRF to
+render the head part, then train a torso-NeRF to render the torso part with the rendering image of the
+head-NeRF as background. Following Guo et al. (2021), we assume the torso part is in canonical
+space and provide the head pose h to torso-NeRF as a signal to infer the torso movement. The
+torso-NeRF implicitly learns to expect the location of the rendered head, then rigid the torso from
+canonical space to render a natural result.
+However, this cooperation between head-NeRF and torso-NeRF is fragile since the torso-NeRF
+cannot observe the head-NeRF’s actual output. Consequently, several recent works report that the
+torso-NeRF produces head-torso separation artifacts Liu et al. (2022)Yao et al. (2022) when the head
+pose is relatively large. Based on the analysis above, we propose to provide the torso-NeRF with a
+perception of the rendering result of the head-NeRF. Specifically, we use the output color Chead of
+the head-NeRF as a pixel-wise condition of the torso-NeRF. The torso’s implicit function Ftorso is
+expressed as:
+Ftorso : (x, Chead; d0, Π, l) → (c, σ)
+(9)
+where d0 is view direction in the canonical space, Π ∈ R3×4 is the head pose that composed of a
+rotation matrix and a transform vector.
+Training Process
+We extract 3D landmarks from the video frames and use these landmark-image
+pairs to train our NeRF-based renderer. The optimization target of head-NeRF and torso-NeRF is
+to reduce the photo-metric reconstruction error between rendered and ground-truth images. Specifi-
+cally, the loss function can be formulated as:
+LNeRF (θ) =
+�
+r∈R
+||Cθ(r, l) − Cg||2
+2
+(10)
+6
+
+Published as a conference paper at ICLR 2023
+where R is the set of camera rays, Cg is the color of the ground image.
+4
+EXPERIMENTS
+4.1
+DATASET PREPARATION AND PREPROCESSING
+Dataset preparation.
+Our method aims to synthesize high-fidelity talking face images with great
+generalizability to out-domain audio. To learn robust audio-to-motion mapping, a large-scale lip-
+reading corpus is needed. Hence we use LRS3-TEDAfouras et al. (2018) to train our variational
+generator and post-net 2. Additionally, a certain person’s speaking video of a few minutes in length
+with an audio track is needed to learn a NeRF-based person portrait renderer. To be specific, in order
+to compare with the state-of-the-art method, we utilize the data set of Lu et al. (2021) and Guo et al.
+(2021), which consist of 5 videos of an average length of 6,000 frames in 25 fps.
+Data preprocessing.
+As for the audio track, we downsample the speech wave into the sampling
+rate of 16000 and process it with a pretrained HuBERT model. For the video frames of LRS3 and
+the target person videos, we resample them into 25 fps and use Deng et al. (2019) to extract the head
+pose and 3D landmarks. As for the target person videos, they are cropped into 512x512 and each
+frame is processed with the help of an automatic parsing method Lee et al. (2020) for segmenting
+the head and torso part and extracting a clean background.
+4.2
+EXPERIMENTAL SETTINGS
+Comparison baselines.
+We compare our GeneFace with several remarkable works: 1) Wav2Lip
+Prajwal et al. (2020), which pretrain a sync-expert to improve the lip-synchronization performance;
+2) MakeItTalk Zhou et al. (2020), which also utilize 3D landmark as the action representation; 3)
+PC-AVS Zhou et al. (2021), which first modularize the audio-visual representation. 4) LiveSpeech-
+Portriat Lu et al. (2021), which achieves photorealistic results at over 30fps; 5) AD-NeRF Guo et al.
+(2021), which first utilize NeRF to achieve talking head generation. For Wav2Lip, PC-AVS, and
+MakeItTalk, the LRS3-TED dataset is used to train the model, and a reference clip of the target
+person video is used during the inference stage; for LSP, both of LRS3-TED dataset and the target
+person video is used to train the model; for the NeRF-based method, AD-NeRF, only the target
+person video is used to train an end-to-end audio-to-image renderer.
+Implementation Details.
+We train the GeneFace on 1 NVIDIA RTX 3090 GPU, and the detailed
+training hyper-parameters of the variational generator, post-net, and NeRF-based are listed in Ap-
+pendix B. For variational generator and post-net, it takes about 40k and 12k steps to converge (about
+12 hours). For the NeRF-based renderer, we train each model for 800k iterations (400k for head and
+400k for the torso, respectively), which takes about 72 hours.
+4.3
+QUANTITATIVE EVALUATION
+Evaluation Metrics
+We employ the FID score Heusel et al. (2017) to measure image quality. We
+utilize the landmark distance (LMD)Chen et al. (2018) and syncnet confidence score Prajwal et al.
+(2020) to evaluate lip synchronization. Furthermore, to evaluate the generalizability, we additionally
+test all methods with a set of out-of-domain (OOD) audio, which consists of cross-lingual, cross-
+gender, and singing voice audios.
+Evaluation Results
+The results are shown in Table 1. We have the following observations. (1) Our
+GeneFace achieves good lip-synchronization with high generalizability. Since Wav2Lip is jointly
+trained with SyncNet, it achieves the highest sync score that is higher than the ground truth video.
+Our method performs best in LMD and achieves a better sync score than other baselines. When
+tested with out-of-domain audios, while the sync-score of person-specific methods (LSP and AD-
+NeRF) significantly drops, GeneFace maintains good performance. (2) Our GeneFace achieves the
+best visual quality. We observe that one-shot methods (Wav2Lip, MakeItTalk, and PC-AVS) perform
+2we select samples of good quality in the LRS3-TED dataset, the selected subset contains 19,775 short
+videos from 3,231 speakers and is about 120 hours-long.
+7
+
+Published as a conference paper at ICLR 2023
+Method
+FID ↓
+LMD↓
+Sync ↑
+FID(OOD) ↓
+Sync(OOD) ↑
+Wav2Lip
+71.40
+3.988
+9.212
+68.05
+9.645
+MakeitTalk
+57.96
+4.848
+4.981
+53.33
+4.933
+PC-AVS
+96.81
+5.812
+6.239
+98.31
+6.156
+LSP
+29.30
+4.589
+6.119
+35.21
+4.320
+AD-NeRF
+27.52
+4.199
+4.894
+35.69
+4.225
+Ground Truth
+0.00
+0.000
+8.733
+N/A
+N/A
+GeneFace (ours)
+22.88
+3.933
+6.987
+27.38
+6.212
+Table 1: Quantitative evaluation with different methods. Best results are in bold.
+/ɔɪ/
+/um/
+/f/
+/um/
+/ʃ/
+GeneFace
+AD-NeRF
+/i/
+/w/
+/s/
+/ɒ/
+/ju:/
+Audio
+Figure 5: The comparison of generated key frame results. We show the phonetic symbol of the
+key frame and the corresponding synthesized talking heads of AD-NeRF and GeneFace. We mark
+the head-torso separation artifact, blurry mouth, un-sync results with brown, blue, and red arrow,
+respectively. Please zoom in for better visualization. More qualitative comparisons can be found
+in demo video.
+poorly on FID due to low image fidelity. Since we use 3D landmarks as the condition of the NeRF
+renderer, it address the mean face problem and leads to better lip syncronization and visual quality
+than AD-NeRF.
+4.4
+QUALITATIVE EVALUATION
+To compare the generated results of each method, we show the keyframes of two clips in Fig.5. Due
+to space limitations, we only compare our GeneFace with AD-NeRF here and provide full results
+with all baselines in Appendix C.1. We observe that although both methods manage to generate
+high-fidelity results, GeneFace solves several problems that AD-NeRF has: 1) head-torso separation
+(brown arrow) due to the separate generation pipeline of head and torso part; 2) blurry mouth images
+due to the one-to-many audio-to-lip mapping; 3) unsynchronized lip due to the weak generalizability.
+User Study
+We conduct user studies to test the quality of audio-driven portraits. Specifically, we
+sample 10 audio clips from English, Chinese, and German for all methods to generate the videos,
+and then involve 20 attendees for user studies. We adopt the Mean Opinion score (MOS) rating
+protocol for evaluation, which is scaled from 1 to 5. The attendees are required to rate the videos
+based on three aspects: (1) lip-sync accuracy; (2) video realness; (3) image quality.
+We compute the average score for each method, and the results are shown in Table 2. We have
+the following observations: 1) Our GeneFace achieves comparatively high lip-sync accuracy with
+Wav2LipPrajwal et al. (2020) since both of them learn a generalized audio-to-motion mapping on a
+large dataset with guidance from a sync-expert. 2) As for the video realness and image quality, the
+Person-specific methods (LSP, AD-NeRF, and GeneFace) outperform one-shot methods (Wav2Lip,
+MakeItTalk, and PC-AVS). Although LSP has slightly better image quality than GeneFace, our
+method achieves the highest video realness and lip-sync accuracy score.
+4.5
+ABLATION STUDY
+In this section, we perform ablation study to prove the necessity of each component in GeneFace.
+8
+
+Published as a conference paper at ICLR 2023
+Methods
+Wav2Lip
+MakeItTalk
+PC-AVS
+LSP
+AD-NeRF
+GeneFace (ours)
+Lip-sync Accuracy
+3.77±0.25
+2.86±0.33
+3.11±0.30
+3.65±0.20
+3.05±0.26
+3.82±0.24
+Image Quality
+3.38±0.19
+2.84±0.20
+2.73±0.25
+3.92±0.13
+3.44±0.22
+3.87±0.16
+Video Realness
+3.27±0.26
+2.52±0.30
+2.46±0.28
+3.62±0.24
+3.31±0.24
+3.87±0.16
+Table 2: User study with different methods. The error bars are 95% confidence interval.
+Setting
+FID↓
+LMD↓
+Sync↑
+FID(OOD)↓
+Sync(OOD)↑
+GeneFace
+22.88
+3.933
+6.987
+27.38
+6.212
+w/o prior flow
+24.71
+4.063
+6.404
+29.55
+5.831
+w/o sync-expert
+24.02
+4.151
+5.972
+30.77
+5.549
+w/o post-net
+30.26
+4.532
+5.085
+35.58
+5.248
+w. fine-tune
+25.75
+4.227
+6.875
+29.30
+5.966
+w/o head-aware
+26.34
+3.948
+6.899
+28.89
+6.167
+Table 3: Ablation study results. The ablation settings are described in Sec. 4.5.
+Varaiational motion generator
+We test two settings on the variational motion generator: (1) w/o
+prior flow, where we replace the flow-based prior with a gaussian prior. The results are shown in
+Table 3 (line 2), where the Sync score drops by a relatively large margin. This observation suggests
+that the temporal enhanced latent variable contributes to the stability of the predicted landmark se-
+quence. (2) w/o sync-expert (line 3), where the variational motion generator is no longer supervised
+by a pretrained sync-expert. We observe that it leads to a significant degradation in Sync score.
+Domain adaptative post-net
+In the setting w/o post-net, we remove the domain adaptative post-
+net, the results are shown in Table 3 (line 4). It can be seen that directly using the 3D landmarks
+predicted by the variational motion generator leads to a significant performance drop in FID and
+Sync scores. To further investigate the efficacy of post-net, we utilize T-SNE to visualize the land-
+marks of different domains in Fig. 10. The visualization results prove that there exists a significant
+domain gap between the LRS3 dataset and the target person video, and our post-net successfully
+rigs the predicted landmarks from the LRS3 domain into the target person domain. We also try to
+replace the post-net with directly fine-tuning on the target person video (line 5), although it achieves
+a competitive sync score on in-domain audios, its performance in OOD audio is worse.
+Head-aware torso-NeRF
+In the w/o head-ware setting, we remove head image condition of the
+torso-NeRF. The results are shown in Table 3 (line 6). Due to the unawareness of the head’s location,
+the head-torso separation occurs occasionally, which results in a drop in the FID score.
+5
+CONCLUSION
+In this paper, we propose GeneFace for talking face generation, which aims to solve the weak gen-
+eralizability and mean face problem faced by previous NeRF-based methods. A variational motion
+generator is proposed to construct a generic audio-to-motion mapping based on a large corpus. We
+then introduce a domain adaptative post-net with an adversarial training pipeline to rig the predicted
+motion representation into the target person domain. Moreover, a head-aware torso-NeRF is present
+to address the head-torso separation issue. Extensive experiments show that our method achieves
+more generalized and high-fidelity talking face generation compared to previous methods. Due to
+space limitations, we discuss the limitations and future work in Appendix D.
+ACKNOWLEDGMENT
+This work was supported in part by the National Natural Science Foundation of China Grant No.
+62222211, Zhejiang Electric Power Co.,Ltd.Science and Technology Project No.5211YF22006 and
+Yiwise.
+9
+
+Published as a conference paper at ICLR 2023
+ETHICS STATEMENT
+GeneFace improves the lip synchronization and expressiveness of the synthesized talking head
+video. With the development of talking face generation techniques, it is much easier for people
+to synthesize fake videos of arbitrary persons. In most situations, they utilize these techniques to
+facilitate the movie and entertainment industry and reduce the bandwidth of video streaming by
+sending audio signals only. However, the talking face generation techniques can be misused. As it is
+more difficult for people to distinguish synthesized videos, the algorithm may be utilized to spread
+fake information or obtain illegal profits. Potential solutions like digital face forensics methods to
+detect deepfakes must be considered. We also plan to include restrictions in the open-source license
+of the GeneFace project to prevent ”deepfake”-related abuse. We hope the public is aware of the
+potential risks of misusing new techniques.
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+Conv1D + ReLU
++Layer Norm
+Non-Causal
+WavNet
+𝜇, 𝜎
+ℎ𝑢𝑏𝑒𝑟𝑡
+𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘
+𝑧~𝑁 𝜇, 𝜎
+(a) Encoder
+Non-Causal
+WavNet
+TransposedConv1D 
++ ReLU +Layer Norm
+ℎ𝑢𝑏𝑒𝑟𝑡
+𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘
+𝑧
+(b) Decoder
+Flip
+Conv1D
+Coupling Layer
+ℎ𝑢𝑏𝑒𝑟𝑡
+x N
+𝑧~𝑁 0,1
+𝑧!
+𝑧"
+(c) Flow-based Prior
+Figure 6: The structure of encoder, decoder, and flow-based prior in variational motion generator.
+Ran Yi, Zipeng Ye, Juyong Zhang, Hujun Bao, and Yong-Jin Liu. Audio-driven talking face video
+generation with learning-based personalized head pose. arXiv preprint arXiv:2002.10137, 2020.
+Lingyun Yu, Jun Yu, Mengyan Li, and Qiang Ling. Multimodal inputs driven talking face gener-
+ation with spatial–temporal dependency. IEEE Transactions on Circuits and Systems for Video
+Technology, 31(1):203–216, 2020.
+Zhimeng Zhang, Lincheng Li, Yu Ding, and Changjie Fan.
+Flow-guided one-shot talking face
+generation with a high-resolution audio-visual dataset. In CVPR, pp. 3661–3670, 2021.
+Hang Zhou, Yu Liu, Ziwei Liu, Ping Luo, and Xiaogang Wang. Talking face generation by adver-
+sarially disentangled audio-visual representation. In AAAI, volume 33, pp. 9299–9306, 2019.
+Hang Zhou, Yasheng Sun, Wayne Wu, Chen Change Loy, Xiaogang Wang, and Ziwei Liu. Pose-
+controllable talking face generation by implicitly modularized audio-visual representation. In
+CVPR, pp. 4176–4186, 2021.
+Yang Zhou, Xintong Han, Eli Shechtman, Jose Echevarria, Evangelos Kalogerakis, and Dingzeyu
+Li. Makelttalk: speaker-aware talking-head animation. ACM Transactions on Graphics (TOG),
+39(6):1–15, 2020.
+A
+DETAILS OF MODELS
+A.1
+VARIATIONAL MOTION GENERATOR
+Following PortaSpeech, our variational motion generator consists of an encoder, a decoder, and
+a flow-based prior model. The encoder, as shown in Fig. 6a, is composed of a 1D-convolution
+followed by ReLU activation and layer normalization, and a non-causal WaveNet. The decoder, as
+shown in Fig. 6b, consists of a non-causal WaveNet and a 1D transposed convolution followed by
+ReLU and layer normalization. The prior model, as shown in Fig. 6c, is a normalizing flow, which is
+composed of a 1D-convolution coupling layer and a channel-wise flip operation. HuBERT features
+are utilized as the audio condition of these three modules.
+A.2
+SYNC-EXPERT
+Our sync-expert inputs a window of Tl consecutive 3D landmark frames and an audio feature clip of
+size Ta × D, where Tl and Ta are the lengths of the video and audio clip respectively, and D is the
+dimension of HuBERT features. The sync-expert is trained to discriminate whether the input audio
+and landmarks are synchronized. It consists of a landmark encoder and an audio encoder, as shown
+in Fig. 7, both of which are comprised of a stack of 1D-convolutions followed by batch normal-
+ization and ReLU. We use cosine-similarity with binary cross-entropy loss to train the sync-expert.
+Specifically, we compute cosine-similarity for the landmark embedding l and audio embedding a
+12
+
+Published as a conference paper at ICLR 2023
+Conv1D+BN
+ReLU
+x N
+ℎ𝑢𝑏𝑒𝑟𝑡 𝑐𝑙𝑖𝑝
+Conv1D+BN
+ReLU
+x N
+𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑐𝑙𝑖𝑝
+cosine 
+similarity
+Pos/Neg?
+Figure 7: The structure of sync-expert.
+Conv1D + ReLU
++Batch Norm
+𝑐𝑜𝑎𝑟𝑠𝑒 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘
+𝑟𝑒𝑓𝑖𝑛𝑒𝑑 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘
++
+x N
+Figure 8: The structure of post-net.
+to represent the probability that the input audio-landmark pair is synchronized. The training loss of
+sync-expert can be represented as:
+Lsync = CE(
+a · l
+max(||a||2 · ||l||2, ϵ))
+(11)
+A.3
+DOMAIN ADAPTATIVE POST-NET AND DISCRIMINATOR
+The domain adaptative post-net, as shown in Fig.
+8, is composed of a stack of residual 1D-
+convolution followed by ReLU and batch normalization. The discriminator is a MLP composed
+of a stack of fully connected layers followed with ReLU and dropout.
+B
+DETAILED EXPERIMENTAL SETTINGS
+B.1
+MODEL CONFIGURATIONS
+We list the hyper-parameters of GeneFace in Tab. 4.
+C
+ADDITIONAL EXPERIMENTS
+C.1
+QUALITATIVE RESULTS WITH ALL BASELINES
+To compare the generated results of each method, we show the keyframes of one in-domain audio
+clip in Fig.9. We have the following observations: 1) Wav2Lip achieves competitive lip-sync perfor-
+mance yet generates blurry mouth results; 2) MakeItTalk and PC-AVS fail to preserve the speaker’s
+identity, leading to unrealistic generated results; 3) LSP generates unnatural lip movement during the
+transition phase of different syllables. Please see our supplementary video for better visualization.
+13
+
+Published as a conference paper at ICLR 2023
+Table 4: Hyper-parameter list
+Hyper-parameter
+GeneFace
+Variational Motion Generator
+Encoder Layers
+8
+Decoder Layers
+4
+Encoder/Decoder Conv1D Kernel
+5
+Encoder/Decoder Conv1D Channel Size
+192
+Latent Size
+16
+Prior Flow Layers
+4
+Prior Flow Conv1D Kernel
+3
+Prior Flow Conv1D Channel Size
+64
+Sync-expert Layers
+14
+Sync-expert Channel Size
+512
+Post-net and Discriminator
+Post-net Layers
+8
+Post-net Conv1D Kernel
+3
+Post-net Conv1D Channel Size
+256
+Discrimnator Layers
+5
+Discrimnator Linear Hidden Size
+256
+Discrimnator Dropout Rate
+0.25
+NeRF-based Renderer
+Head/Torso-NeRF Layers
+11
+Head/Torso-NeRF Hidden Size
+256
+Landmark/Head Color Encoder Layers
+3
+Landmark/Head Color Encoder Hidden Size
+128
+Setting
+L2 error on 3D landmark↓
+LMD↓
+GeneFace (VAE + Flow + landmark NeRF)
+0.0371
+3.933
+vanilla VAE + landmark NeRF
+0.0385
+4.063
+Regression Model + landmark NeRF
+0.0424
+4.305
+AD-NeRF
+N/A
+4.199
+Table 5: Ablation study on 3D Landmark L2 error.
+C.2
+EVALUATION ON 3D LANDMARK L2 ERROR
+To evaluate the contribution of the variational generator to the quality of the predicted landmark, we
+adopt L2 error on the predicted 3D landmarks as the metric. We compare our vairiaitonal generator
+(VAE+Flow) against vanilla VAE and a simple regression model trained with MSE loss. The results
+are listed in Table 5. It can be seen that removing the prior flow or using a regression-based model
+leads to a performance drop.
+C.3
+T-SNE VISUALIZATION FOR DOMAIN ADAPTATION
+To further investigate the efficacy of post-net, we utilize T-SNE to visualize the landmarks of dif-
+ferent domains in Fig. 10. The visualization results prove that there exists a significant domain
+gap between the LRS3 dataset and the target person video, and our post-net successfully rigs the
+predicted landmarks from the LRS3 domain into the target person domain.
+D
+LIMITATIONS AND FUTURE WORK
+There are mainly two limitations of the proposed approach. Firstly, we found the landmark sequence
+generated by variational motion generator and post-net occasionally has tiny fluctuations, which
+results in some artifacts such as shaking hairs, etc. Currently, we utilize a heuristic post-processing
+method (Gaussian filter) to alleviate this problem. In future work, we will explore better modeling
+the temporal information in the network architecture to further improve the stability. Secondly, the
+current NeRF-based renderer is majorly based on the setting of vanilla NeRF, which results in a long
+training and inference time. In future work, we will try to enhance the performance of the NeRF
+backend by combining recent progress in accelerated and light-weight NeRF.
+14
+
+Published as a conference paper at ICLR 2023
+GeneFace
+AD-NeRF
+GT
+LSP
+PC-AVS
+MakeItTalk
+Wav2Lip
+/i/
+/w/
+/s/
+/ɒ/
+/ju:/
+Audio
+Figure 9: The comparison of generated key frame results. We show the phonetic symbol of the
+key frame and the corresponding synthesized talking heads of all baselines. Please zoom in for
+better visualization. More qualitative comparisons can be found in demo video.
+Figure 10: The T-SNE visualization of 3DMM landmarks in different datasets. The green and blue
+points denote the ground truth landmarks in LRS3 dataset and the target person video; The red and
+yellow points represent the predicted landmarks without/with the domain adaptation.
+15
+
+person_train
+postnet
+pred_Irs3
+vae

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+arXiv:2301.02632v1  [math.GM]  30 Nov 2022
+A NOTE ON LP-KENMOTSU MANIFOLDS ADMITTING
+RICCI-YAMABE SOLITONS
+MOBIN AHMAD, GAZALA AND MOHD BILAL
+Abstract. In the current note, we study Lorentzian para-Kenmotsu (in brief,
+LP -Kenmotsu) manifolds admitting Ricci-Yamabe solitons (RYS) and gradi-
+ent Ricci-Yamabe soliton (gradient RYS). At last by constructing a 5-dimensional
+non-trivial example we illustrate our result.
+2010 Mathematics Subject Classification. 53C20, 53C21, 53C25, 53E20.
+Keywords. Lorentzian para-Kenmotsu manifolds, Ricci-Yamabe solitons, Einstein
+manifolds, ν-Einstein manifolds.
+1. Introduction
+In 2019, a scalar combination of Ricci and Yamabe flows was proposed by the
+authors G¨uler and Crasmareanu [6], this advanced class of geometric flows called
+Ricci-Yamabe (RY) flow of type (σ, ρ) and is defined by
+∂
+∂tg(t) + 2σS(g(t)) + ρr(t)g(t) = 0,
+g(0) = g0
+for some scalars σ and ρ.
+A solution to the RY flow is called RYS if it depends only on one parameter group
+of diffeomorphism and scaling. A Riemannian (or semi-Riemannian) manifold M
+is said to have a RYS if
+£Kg + 2σS + (2Λ − ρr)g = 0,
+(1.1)
+where σ, ρ, Λ ∈ R (the set of real numbers).
+If K is the gradient of a smooth
+function v on M, then (1.1) is called the gradient Ricci-Yamabe soliton (gradient
+RYS) and hence (1.1) turns to
+∇2v + σS + (Λ − ρr
+2 )g = 0,
+(1.2)
+where ∇2v is the Hessian of v. It is to be noted that a RYS of types (σ, 0) and
+(0, ρ) are known as σ−Ricci soliton and ρ−Yamabe soliton, respectively. A RYS
+is said to be shrinking , steady or expanding if Λ < 0, = 0 or > 0, respectively. A
+RYS is said to be a
+• Ricci soliton [7] if σ = 1, ρ = 0,
+• Yamabe soliton [8] if σ = 0, ρ = 1,
+• Einstein soliton [3] if σ = 1, ρ = −1,
+As a continuation of this study, we tried to study RYS in the frame-work of
+LP-Kenmotsu manifolds of dimension n. We recommend the papers [1, 2, 5, 9, 10,
+13, 15, 16, 17, 18, 19] and the references therein for more details about the related
+studies.
+1
+
+2
+MOBIN AHMAD, GAZALA AND MOHD BILAL
+2. Preliminaries
+An n-dimensional differentiable manifold M with structure (ϕ, ζ, ν, g) is said to
+be a Lorentzian almost paracontact metric manifold, if it admits a (1, 1)-tensor field
+ϕ, a contravariant vector field ζ, a 1-form ν and a Lorentzian metric g satisfying
+(2.1)
+ν(ζ) + 1 = 0,
+(2.2)
+ϕ2E = E + ν(E)ζ,
+(2.3)
+ϕζ = 0,
+ν(ϕE) = 0,
+(2.4)
+g(ϕE, ϕF) = g(E, F) + ν(E)ν(F),
+(2.5)
+g(E, ζ) = ν(E),
+(2.6)
+ϕ(E, F) = ϕ(F, E) = g(E, ϕF)
+for any vector fields E, F ∈ χ(M), where χ(M) is the Lie algebra of vector fields
+on M.
+If ζ is a killing vector field, the (para) contact structure is called a K-(para) contact.
+In such a case, we have
+(2.7)
+∇Eζ = ϕE.
+Recently, the authors Haseeb and Prasad defined and studied the following notion:
+Definition 2.1. A Lorentzian almost paracontact manifold M is called Lorentzian
+para-Kenmostu manifold if [11]
+(2.8)
+(∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE
+for any E, F on M.
+In an LP-Kenmostu manifold, we have
+(2.9)
+∇Eζ = −E − ν(E)ζ,
+(2.10)
+(∇Eν)F = −g(E, F) − ν(E)ν(F),
+where ∇ denotes the Levi-Civita connection respecting to the Lorentzian metric g.
+Furthermore, in an LP-Kenmotsu manifold, the following relations hold [11]:
+(2.11)
+g(R(E, F)G, ζ) = ν(R(E, F)G) = g(F, G)ν(E) − g(E, G)ν(F),
+(2.12)
+R(ζ, E)F = −R(E, ζ)F = g(E, F)ζ − ν(F)E,
+(2.13)
+R(E, F)ζ = ν(F)E − ν(E)F,
+(2.14)
+R(ζ, E)ζ = E + ν(E)ζ,
+(2.15)
+S(E, ζ) = (n − 1)ν(E), S(ζ, ζ) = −(n − 1),
+(2.16)
+Qζ = (n − 1)ζ
+for any E, F, G ∈ χ(M), where R, S and Q represent the curvature tensor, the Ricci
+tensor and the Q Ricci operator, respectively.
+
+A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS
+3
+Definition 2.2. [21] An LP-Kenmotsu manifold M is said to be ν-Einstein man-
+ifold if its S(̸= 0) is of the form
+(2.17)
+S(E, F) = ag(E, F) + bν(E)ν(F),
+where a and b are smooth functions on M. In particular, if b = 0, then M is termed
+as an Einstein manifold.
+Remark 2.3. [12] In an LP-Kenmotsu manifold of n-dimension, S is of the form
+(2.18)
+S(E, F) = (
+r
+n − 1 − 1)g(E, F) + (
+r
+n − 1 − n)ν(E)ν(F),
+where r is the scalar curvature of the manifold.
+Lemma 2.4. In an n-dimensional LP-Kenmotsu manifold, we have
+(2.19)
+ζ(r) = 2(r − n(n − 1)),
+(2.20)
+(∇EQ)ζ = QE − (n − 1)E,
+(2.21)
+(∇ζQ)E = 2QE − 2(n − 1)E
+for any E on M.
+Proof. Equation (2.18) yields
+(2.22)
+QE = (
+r
+n − 1 − 1)E + (
+r
+n − 1 − n)ν(E)ζ.
+Taking the covariant derivative of (2.22) with respect to F and making use of (2.9)
+and (2.10), we lead to
+(∇F Q)E = F(r)
+n − 1(E + ν(E)ζ) − (
+r
+n − 1 − n)(g(E, F)ζ + ν(E)F + 2ν(E)ν(F)ζ).
+By contracting F in the foregoing equation and using trace {F → (∇F Q)E} =
+1
+2E(r), we find
+n − 3
+2(n − 1)E(r) =
+� ζ(r)
+n − 1 − (r − n(n − 1))
+�
+ν(E),
+which by replacing E by ζ and using (2.1) gives (2.19). We refer the readers to see
+[14] for the proof of (2.20) and (2.21).
+□
+Remark 2.5. From the equation (2.19), it is noticed that if an n-dimensional
+LP-Kenmotsu manifold possesses the constant scalar curvature, then r = n(n − 1)
+and hence (2.18) reduces to S(E, F) = (n − 1)g(E, F). Thus, the manifold under
+consideration is an Einstein manifold.
+3. Ricci-Yamabe solitons on LP-Kenmotsu manifolds
+Let the metric of an n-dimensional LP-Kenmotsu manifold be a Ricci-Yamabe
+soliton (g, K, Λ, σ, ρ), then (1.1) holds. By differentiating (1.1) covariantly with
+resprct to G, we have
+(∇G£Kg)(E, F)
+=
+−2σ(∇GS)(E, F) + ρ(Gr)g(E, F).
+(3.1)
+Since ∇g = 0, then the following formula [20]
+(£K∇Eg −∇E£Kg −∇[K,E]g)(F, G) = −g((£K∇)(E, F), G)−g((£K∇)(E, G), F)
+
+4
+MOBIN AHMAD, GAZALA AND MOHD BILAL
+turns to
+(∇E£Kg)(F, G) = g((£K∇)(E, F), G) + g((£K∇)(E, G), F).
+Since the operator £K∇ is symmetric, therefore we have
+2g((£K∇)(E, F), G) = (∇E£Kg)(F, G) + (∇F £Kg)(E, G) − (∇G£Kg)(E, F),
+which by using (3.1) takes the form
+2g((£K∇)(E, F), G)
+=
+−2σ[(∇ES)(F, G) + (∇F S)(G, E) + (∇GS)(E, F)]
++ρ[(Er)g(F, G) + (Fr)g(G, E) + (Gr)g(E, F)].
+(3.2)
+Putting F = ζ in (3.2) and using (2.5), we find
+2g((£K∇)(E, ζ), G)
+=
+−2σ[(∇ES)(ζ, G) + (∇ζS)(G, E) − (∇GS)(E, ζ)]
++ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)]
+(3.3)
+By virtue of (2.20) and (2.21), (3.3) leads to
+2g((£K∇)(E, ζ), G)
+=
+−4σ[S(E, G) − (n − 1)g(E, G)]
++ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)].
+By eliminating G from the foregoing equation, we have
+2(£K∇)(F, ζ)
+=
+ρg(Dr, F)ζ − ρ(Dr)ν(F) − 4σQF
+(3.4)
++[4σ(n − 1) + 2ρ(r − n(n − 1))]F.
+If we take r as constant, then from (2.19) we find r = n(n − 1), and hence (3.4)
+reduces to
+(£K∇)(F, ζ)
+=
+−2σQF + 2σ(n − 1)F.
+(3.5)
+Taking covariant derivative of (3.5) with respect to E, we have
+(∇E£K∇)(F, ζ)
+=
+(£K∇)(F, E) − 2σν(E)[QF − (n − 1)F]
+(3.6)
+−
+2σ(∇EQ)F.
+Again from [20], we have
+(£KR)(E, F)G = (∇E£K∇)(F, G) − (∇F £K∇)(E, G),
+which by putting G = ζ and using (3.6) takes the form
+(£KR)(E, F)ζ
+=
+2σν(F)(QE − (n − 1)E) − 2σν(E)(QF − (n − 1)F)
+(3.7)
+−2σ((∇EQ)F − (∇F Q)E).
+Putting F = ζ in (3.7) then using (2.1), (2.2), (2.20) and (2.21), we arrive at
+(£KR)(E, ζ)ζ = 0.
+(3.8)
+The Lie derivative of R(E, ζ)ζ = −E − ν(E)ζ along K leads to
+(£KR)(E, ζ)ζ − g(E, £Kζ)ζ + 2ν(£Kζ)E = −(£Kν)(E)ζ.
+(3.9)
+From (3.8) and (3.9), we have
+(£Kν)(E)ζ = −2ν(£Kζ)E + g(E, £Kζ)ζ.
+(3.10)
+Taking the Lie derivative of g(E, ζ) = ν(E), we find
+(£Kν)(E) = g(E, £Kζ) + (£Kg)(E, ζ).
+(3.11)
+By putting F = ζ in (1.1) and using (2.15), we have
+(£Kg)(E, ζ) = −{2σ(n − 1) + 2Λ − ρn(n − 1)}ν(E),
+(3.12)
+
+A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS
+5
+where r = n(n − 1) being used.
+The Lie derivative of g(ζ, ζ) = −1 along K we lead to
+(£Kg)(ζ, ζ) = −2ν(£Kζ).
+(3.13)
+From (3.12) and (3.16), we find
+ν(£Kζ) = −{σ(n − 1) + Λ − ρn(n − 1)
+2
+}.
+(3.14)
+Now, combining the equations (3.10), (3.11), (3.12) and (3.17), we find
+Λ = ρn(n − 1)
+2
+− σ(n − 1).
+(3.15)
+Thus, we have
+Theorem 3.1. Let (M, g) be an n-dimensional LP-Kenmotsu manifold admitting
+Ricci-Yamabe soliton (g, K, Λ, σ, ρ) with constant scalar curvature tensor, then Λ =
+ρn(n−1)
+2
+− σ(n − 1).
+For σ = 1 and ρ = 0, from (3.15) we have Λ = −(n − 1). Thus, we have the
+following:
+Corollary 3.2. If an n-dimensional LP-Kenmotsu manifold admits a Ricci soliton
+with constant scalar curvature, then the soliton is shrinking.
+For σ = 0 and ρ = 1, from (3.15) we have Λ =
+n(n−1)
+2
+. Thus, we have the
+following:
+Corollary 3.3. If an n-dimensional LP-Kenmotsu manifold admits a Yamabe
+soliton with constant scalar curvature, then the soliton is shrinking.
+For σ = 1 and ρ = −1, from (3.15) we have Λ = − (n2−1)
+2
+. Thus, we have the
+following:
+Corollary 3.4. If an n-dimensional LP-Kenmotsu manifold admits an Einstein
+soliton with constant scalar curvature, then the soliton is shrinking.
+Now, we consider the metric of an n-dimensional LP-Kenmotsu manifold as a
+Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ), then from (1.1) and (2.9) we have
+S(E, F) = − 1
+σ (Λ − 1 − ρr
+2 )g(E, F) + 1
+σ ν(E)ν(F),
+where σ ̸= 0.
+(3.16)
+By putting F = ζ in (3.16) and using (2.15), we find
+Λ = ρr
+2 − σ(n − 1).
+(3.17)
+Now, comparing (2.18) and (3.17), we have r = n−1
+σ
++ n(n − 1), which by using in
+(3.17) it follows that Λ = −σ(n − 1) + ρ(n−1)(1+nσ)
+2σ
+. Thus, we have the following
+theorem:
+Theorem 3.5. An n-dimensional LP-Kenmotsu manifold with constant scalar
+curvature admitting Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ) is an ν-Einstein manifold.
+Moreover, the soliton is expanding, steady or shrinking according to ρ
+σ > 2σ − ρn,
+ρ
+σ = 2σ �� ρn, or ρ
+σ < 2σ − ρn.
+
+6
+MOBIN AHMAD, GAZALA AND MOHD BILAL
+4. Gradient Ricci-Yamabe solitons on LP-Kenmotsu manifolds
+Definition 4.1. A Riemannian (or semi-Riemannian) metric g on M is called a
+gradient RYS, if
+Hessv + σS + (Λ − ρr
+2 )g = 0,
+(4.1)
+where Hessv denotes the Hessian of a smooth function v on M and defined by
+Hessv = ∇∇v.
+Let M be an n-dimensional LP-Kenmotsu manifold with g as a gradient RYS.
+Then equation (4.1) can be written as
+∇EDv + σQE + (Λ − ρr
+2 )E = 0,
+(4.2)
+for all vector fields E on M, where D denotes the gradient operator of g. Taking
+the covariant derivative of (4.2) with respect to F, we have
+∇F ∇EDv = −σ{(∇F Q)E + Q(∇F E)} + ρF(r)
+2
+E − (Λ − ρr
+2 )∇F E.
+(4.3)
+Interchanging E and F in (4.3), we lead to
+∇E∇F Dv = −σ{(∇EQ)F + Q(∇EF)} + ρE(r)
+2
+F − (Λ − ρr
+2 )∇EF.
+(4.4)
+By making use of (4.2)-(4.4), we find
+R(E, F)Dv = σ{(∇F Q)E − (∇EQ)F} + ρ
+2{E(r)F − F(r)E}.
+(4.5)
+Now, from (2.18), we find
+QE = (
+r
+n − 1 − 1)E + (
+r
+n − 1 − n)ν(E)ζ,
+which on taking covariant derivative with repect to F leads to
+(∇F Q)E
+=
+F(r)
+n − 1(E + ν(E)ζ) − (
+r
+n − 1 − n)(g(E, F)ζ
+(4.6)
++2ν(E)ν(F)ζ + ν(E)F).
+By using (4.6) in (4.5), we have
+R(E, F)Dv
+=
+(n − 1)ρ − 2σ
+2(n − 1)
+{E(r)F − F(r)E} +
+σ
+n − 1{F(r)ν(E)ζ − E(r)ν(F)ζ}
+−σ(
+r
+n − 1 − n)(ν(E)F − ν(F)E).
+(4.7)
+Contracting forgoing equation along E gives
+S(F, Dv)
+=
+�(n − 1)2ρ − 2σ(n − 2)
+n − 1
+�
+F(r)
+(4.8)
++σ(n − 3)(r − n(n − 1))
+n − 1
+ν(F).
+From the equation (2.18), we can write
+S(F, Dv) = (
+r
+n − 1 − 1)F(v) + (
+r
+n − 1 − n)ν(F)ζ(v).
+(4.9)
+
+A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS
+7
+Now, by equating (4.8) and (4.9), then putting F = ζ and using (2.1), (2.19), we
+find
+ζ(v) = r − n(n − 1)
+n − 1
+{2(n − 1)ρ − σ(5n − 13)
+n − 1
+}.
+(4.10)
+Taking the inner product of (4.7) with ζ, we get
+F(v)ν(E) − E(v)ν(F) = ρ
+2{E(r)ν(F) − F(r)ν(E)},
+which by replacing E by ζ and using (2.19), (4.10), we infer
+F(v) = −(r − n(n − 1)){3ρ − σ(5n − 13)
+(n − 1)2 }ν(F) − ρ
+2F(r).
+(4.11)
+If we take r as constant, then from Remark 2.5, we get r = n(n − 1). Thus, (4.11)
+leads to F(v) = 0. This implies that v is constant. Thus, the soliton under the
+consideration is trivial. Hence we state:
+Theorem 4.2. If the metric of an LP-Kenmotsu manifold of constant scalar curva-
+ture tensor admitting a special type of vector field is gradient RYS, then the soliton
+is trivial.
+For v constant, (1.2) turns to
+σQE = −(Λ − ρr
+2 )E,
+which leads to
+S(E, F) = − 1
+σ (Λ − ρn(n − 1)
+2
+)g(E, F),
+σ ̸= 0.
+(4.12)
+By putting E = F = ζ in (4.12) and using (2.15), we obtain
+Λ = ρn(n − 1)
+2
+− σ(n − 1).
+(4.13)
+Corollary 4.3. If an n-dimensional LP-Kenmotsu manifold admits a gradient
+Ricci soliton with the constant scalar curvature, then the manifold under the con-
+sideration is an Einstein manifold and Λ = ρn(n−1)
+2
+− σ(n − 1).
+For σ = 1 and ρ = 0, from (4.13) we find Λ = −(n − 1). Thu, we have the
+following:
+Corollary 4.4. If an n-dimensional LP-Kenmotsu manifold admits a gradient
+Ricci soliton with the constant scalar curvature, then the soliton is shrinking.
+For σ = 1 and ρ = −1, from (4.13) we have Λ = − (n−1)(n+2)
+2
+. Thus, we have
+the following:
+Corollary 4.5. If an n-dimensional LP-Kenmotsu manifold admits an gradient
+Einstein soliton with constant scalar curvature, then the soliton is shrinking.
+Example. We consider the 5-dimensional manifold M 5 =
+�
+(x1, x2, x3, x4, x5) ∈ R5 : x5 > 0
+�
+,
+where (x1, x2, x3, x4, x5) are the standard coordinates in R5. Let ̺1, ̺2, ̺3, ̺4 and
+̺5 be the vector fields on M 5 given by
+̺1 = ex5 ∂
+∂x1
+, ̺2 = ex5 ∂
+∂x2
+, ̺3 = ex5 ∂
+∂x3
+, ̺4 = ex5 ∂
+∂x4
+, ̺5 =
+∂
+∂x5
+= ζ,
+
+8
+MOBIN AHMAD, GAZALA AND MOHD BILAL
+which are linearly independent at each point of M 5. Let g be the Lorentzian metric
+defined by
+g(̺i, ̺i) = 1,
+for
+1 ≤ i ≤ 4
+and
+g(̺5, ̺5) = −1,
+g(̺i, ̺j) = 0,
+for
+i ̸= j,
+1 ≤ i, j ≤ 5.
+Let ν be the 1-form defined by ν(E) = g(E, ̺5) = g(̺, ζ) for all E ∈ χ(M 5), and
+let ϕ be the (1, 1)-tensor field defined by
+ϕ̺1 = −̺2, ϕ̺2 = −̺1, ϕ̺3 = −̺4, ϕ̺4 = −̺3, ϕ̺5 = 0.
+By applying linearity of ϕ and g, we have
+ν(ζ) = g(ζ, ζ) = −1, ϕ2E = E + ν(E)ζ and g(ϕE, ϕF) = g(E, F) + ν(E)ν(F)
+for all E, F ∈ χ(M 5). Thus for ̺5 = ζ, the structure (ϕ, ζ, ν, g) defines a Lorentzian
+almost paracontact metric structure on M 5. Then we have
+[̺i, ̺j] = −̺i,
+for
+1 ≤ i ≤ 4, j = 5,
+[̺i, ̺j] = 0,
+otherwise.
+By using Koszul’s formula, we can easily find we obtain
+∇̺i̺j =
+
+
+
+
+
+−̺5,
+1 ≤ i = j ≤ 4,
+−̺i,
+1 ≤ i ≤ 4, j = 5,
+0,
+otherwise.
+Also one can easily verify that
+∇Eζ = −E − η(E)ζ
+and
+(∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE.
+Therefore, the manifold is an LP-Kenmotsu manifold.
+From the above results, we can easily obtain the non-vanishing components of R
+as follows:
+R(̺1, ̺2)̺1 = −̺2, R(̺1, ̺2)̺2 = ̺1, R(̺1, ̺3)̺1 = −̺3, R(̺1, ̺3)̺3 = ̺1,
+R(̺1, ̺4)̺1 = −v4, R(̺1, ̺4)̺4 = ̺1, R(̺1, ̺5)̺1 = −̺5, R(̺1, ̺5)̺5 = −̺1,
+R(̺2, ̺3)̺2 = −̺3, R(̺2, ̺3)̺3 = ̺2, R(̺2, ̺4)̺2 = −̺4, R(̺2, ̺4)̺4 = ̺2,
+R(̺2, ̺5)̺2 = −̺5, R(̺2, ̺5)̺5 = −̺2, R(̺3, ̺4)̺3 = −̺4, R(̺3, ̺4)̺4 = ̺3,
+R(̺3, ̺5)̺3 = −̺5, R(̺3, ̺5)̺5 = −̺3, R(̺4, ̺5)̺4 = −̺5, R(̺4, ̺5)̺5 = −̺4.
+Also, we calculate the Ricci tensors as follows:
+S(̺1, ̺1) = S(̺2, ̺2) = S(̺3, ̺3) = S(̺4, ̺4) = 4,
+S(̺5, ̺5) = −4.
+Therefore, we have
+r = S(̺1, ̺1) + S(̺2, ̺2) + S(̺3, ̺3) + S(̺4, ̺4) − S(̺5, ̺5) = 20.
+Now by taking Dv = (̺1v)̺1 + (̺2v)̺2 + (̺3v)̺3 + (̺4v)̺4 + (̺5v)̺5, we have
+∇̺1Dv = (̺1(̺1v) − (̺5v))̺1 + (̺1(̺2v))̺2 + (̺1(̺3v))̺3 + (̺1(̺4v))̺4 + (̺1(̺5v) − (̺1v))̺5,
+∇̺2Dv = (̺2(̺1v))̺1 + (̺2(̺2v) − (̺5v))̺2 + (̺2(̺3v))̺3 + (̺2(̺4v))̺4 + (̺2(̺5v) − (̺2v))̺5,
+∇̺3Dv = (̺3(̺1v))̺1 + (̺3(̺2v))̺2 + (̺3(̺3v) − (̺5v))̺3 + (̺3(̺4v))̺4 + (̺3(̺5v) − (̺3v))̺5,
+∇̺4Dv = (̺4(̺1v))̺1 + (̺4(̺2v))̺2 + (̺4(̺3v))̺3 + (̺4(̺4v) − (̺5v))̺4 + (̺4(̺5v) − (̺4v))̺5,
+∇̺5Dv = (̺5(̺1v))̺1 + (̺5(̺2v))̺2 + (̺5(̺3v))̺3 + (̺5(̺4v))̺4 + (̺5(̺5v))̺5.
+
+A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS
+9
+Thus, by virtue of (4.2), we obtain
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+̺1(̺1v) − ̺5v = −(Λ + 4σ − 10ρ),
+̺2(̺2v) − ̺5v = −(Λ + 4σ − 10ρ),
+̺3(̺3v) − ̺5v = −(Λ + 4σ − 10ρ),
+̺4(̺4v) − ̺5v = −(Λ + 4σ − 10ρ),
+̺5(̺5v) = −(Λ + 4σ − 10ρ),
+̺1(̺2v) = ̺1(̺3v) = ̺1(̺4v) = 0,
+̺2(̺1v) = ̺2(̺3v) = ̺2(̺4v) = 0,
+̺3(̺1v) = ̺3(̺2v) = ̺3(̺4v) = 0,
+̺4(̺1v) = ̺4(̺2v) = ̺4(̺3v) = 0,
+̺1(̺5v) − (̺1v) = ̺2(̺5v) − (̺2v) = 0,
+̺3(̺5v) − (̺3v) = ̺4(̺5v) − (̺4v) = 0.
+(4.14)
+Thus, the equations in (4.14) are respectively amounting to
+e2x5 ∂2v
+∂x2
+1
+− ∂v
+∂x5
+= −(Λ + 4σ − 10ρ),
+e2x5 ∂2v
+∂x2
+2
+− ∂v
+∂x5
+= −(Λ + 4σ − 10ρ),
+e2x5 ∂2v
+∂x2
+3
+− ∂v
+∂x5
+= −(Λ + 4σ − 10ρ),
+e2x5 ∂2v
+∂x2
+4
+− ∂v
+∂x5
+= −(Λ + 4σ − 10ρ),
+∂2v
+∂x2
+5
+= −(Λ + 4σ − 10ρ),
+∂2v
+∂x1∂x2
+=
+∂2v
+∂x1∂x3
+=
+∂2v
+∂x1∂x4
+=
+∂2v
+∂x2∂x3
+=
+∂2v
+∂x2∂x4
+=
+∂2v
+∂x3∂x4
+= 0,
+ex5
+∂2v
+∂x5∂x1
++ ∂v
+∂x1
+= ex5
+∂2v
+∂x5∂x2
++ ∂v
+∂x2
+= ex5
+∂2v
+∂x5∂x3
++ ∂v
+∂x3
+= ex5
+∂2v
+∂x5∂x4
++ ∂v
+∂x4
+= 0.
+From the above equations it is observed that v is constant for Λ = −4σ + 10ρ.
+Hence, equation (4.2) is satisfied. Thus, g is a gradient RYS with the soliton vector
+field K = Dv, where v is constant and Λ = −4σ + 10ρ. Hence, Theorem 4.2 is
+verified.
+References
+[1] Blaga, A. M., Solitons and geometrical structure in a perfect fluid spacetime, Rocky Mt. J.
+Math. (2020).
+[2] Blaga, A. M., Some geometrical aspects of Einstein, Ricci and Yamabe solitons, J. Geom.
+Symmetry Phys., 52 (2019), 17-26.
+[3] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal., 132(2016), 66-94.
+[4] Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C. and Mazzieri, L., The Ricci Bour-
+guignon flow, Pacific J. Math., 28(2017), 337-370.
+
+10
+MOBIN AHMAD, GAZALA AND MOHD BILAL
+[5] Chidananda,
+S.,
+and
+Venkatesha,
+V.,
+Yamabe
+soliton
+and
+Riemann
+soli-
+ton
+on
+Lorentzian
+para-Sasakian
+manifold,
+Commun.
+Korean
+Math.
+Soc.,
+https://doi.org/10.4134/CKMS.c200365.
+[6] G¨uler, S. and Crasmareanu, M., Ricci-Yamabe maps for Riemannian flows and their volume
+variation and volume entropy, Turk. J. Math., 43 (2019), 2631-2641.
+[7] Hamilton, R. S., Lectures on Geometric Flows (Unpublished manuscript, 1989).
+[8] Hamilton, R. S., The Ricci Flow on Surfaces, Mathematics and General Relativity (Santa
+Cruz, CA, 1986), Contemp. Math., A.M.S., 71 (1988), 237-262.
+[9] Haseeb, A. and De, U. C., η-Ricci solitons in ǫ-Kenmotsu manifolds, J. Geom. 110, 34 (2019).
+[10] Haseeb, A. and Almusawa, H., Some results on Lorentzian para-Kenmotsu manifolds admit-
+ting η-Ricci solitons, Palestine Journal of Mathematics, 11(2)(2022), 205-213.
+[11] Haseeb, A. and Prasad, R., Certain results on Lorentzian para-Kenmotsu manifolds, Bol.
+Soc. Parana. Mat., 39(3) (2021), 201-220.
+[12] Haseeb, A. and Prasad, R., Some results on Lorentzian para-Kenmotsu manifolds, Bull.
+Transilvania Univ. of Brasov, 13(62) (2020), no. 1, 185-198.
+[13] Haseeb, A., Prasad, R. and Mofarreh, F., Sasakian manifolds admitting ∗-η-Ricci-Yamabe
+solitons, Advances in Mathematical Physics, Vol. 2022, Article ID 5718736, 7 pages. doi:
+https:// doi.org/10.1155/2022/5718736
+[14] Li, Y., Haseeb, A. and Ali, M., LP -Kenmotsu manifolds admitting η-Ricci solitons and
+spacetime, Journal of Mathematics, 2022, Article ID 6605127, 10 pages.
+[15] Lone, M. A. and Harry, I. F., Ricci Solitons on Lorentz-Sasakian space forms, Journal of
+Geometry and Physics, 104547, doi: https://doi.org/10.1016/j.geomphys.2022.104547.
+[16] Pankaj, Chaubey, S. K and Prasad, R., Three dimensional Lorentzian para-Kenmotsu mani-
+folds and Yamabe soliton, Honam Mathematical J., 43(4) (2021), 613-626.
+[17] Singh, J. P. and Khatri, M., On Ricci-Yamabe soliton and geometrical structure in a perfect
+fluid spacetime, Afr. Mat., 32(2021), 1645-1656.
+[18] Venkatesha, Kumara, H. A., Ricci soliton and geometrical structure in a perfect fluid space-
+time with torse-forming vector field, Afr. Mat. 30 (2019), 725-736
+[19] Yoldas, H. I., On Kenmotsu manifolds admitting η-Ricci-Yamabe solitons,Int. J. Geom. Met.
+Mod. Phy., 18(12) (2021), 2150189.
+[20] Yano, K., Integral Formulas in Riemannian geometry, Pure and Applied Mathematics, Vol.
+I, Marcel Dekker, New York, 1970.
+[21] Yano, K. and Kon, M., Structures on manifolds, World Scientific, (1984).
+Mobin Ahmad
+Department of Mathematics,
+Integral University, Kursi Road,
+Lucknow-226026.
+Email : [email protected]
+Gazala
+Department of Mathematics,
+Integral University, Kursi Road,
+Lucknow-226026.
+Email : [email protected]
+Mohd. Bilal
+Department of Mathematical Sciences,
+Umm Ul Qura University,
+Makkah, Saudi Arabia.
+Email: [email protected]
+

8dE0T4oBgHgl3EQfwgEX/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf,len=420
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='02632v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='GM]  30 Nov 2022  A NOTE ON LP-KENMOTSU MANIFOLDS ADMITTING  RICCI-YAMABE SOLITONS  MOBIN AHMAD, GAZALA AND MOHD BILAL  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' In the current note, we study Lorentzian para-Kenmotsu (in brief,  LP -Kenmotsu) manifolds admitting Ricci-Yamabe solitons (RYS) and gradi-  ent Ricci-Yamabe soliton (gradient RYS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' At last by constructing a 5-dimensional  non-trivial example we illustrate our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' 53C20, 53C21, 53C25, 53E20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Lorentzian para-Kenmotsu manifolds, Ricci-Yamabe solitons, Einstein  manifolds, ν-Einstein manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Introduction  In 2019, a scalar combination of Ricci and Yamabe flows was proposed by the  authors G¨uler and Crasmareanu [6], this advanced class of geometric flows called  Ricci-Yamabe (RY) flow of type (σ, ρ) and is defined by  ∂  ∂tg(t) + 2σS(g(t)) + ρr(t)g(t) = 0,  g(0) = g0  for some scalars σ and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  A solution to the RY flow is called RYS if it depends only on one parameter group  of diffeomorphism and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' A Riemannian (or semi-Riemannian) manifold M  is said to have a RYS if  £Kg + 2σS + (2Λ − ρr)g = 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1)  where σ, ρ, Λ ∈ R (the set of real numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  If K is the gradient of a smooth  function v on M, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) is called the gradient Ricci-Yamabe soliton (gradient  RYS) and hence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) turns to  ∇2v + σS + (Λ − ρr  2 )g = 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2)  where ∇2v is the Hessian of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' It is to be noted that a RYS of types (σ, 0) and  (0, ρ) are known as σ−Ricci soliton and ρ−Yamabe soliton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' A RYS  is said to be shrinking , steady or expanding if Λ < 0, = 0 or > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' A  RYS is said to be a  Ricci soliton [7] if σ = 1, ρ = 0,  Yamabe soliton [8] if σ = 0, ρ = 1,  Einstein soliton [3] if σ = 1, ρ = −1,  As a continuation of this study, we tried to study RYS in the frame-work of  LP-Kenmotsu manifolds of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' We recommend the papers [1, 2, 5, 9, 10,  13, 15, 16, 17, 18, 19] and the references therein for more details about the related  studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  1  2  MOBIN AHMAD, GAZALA AND MOHD BILAL  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Preliminaries  An n-dimensional differentiable manifold M with structure (ϕ, ζ, ν, g) is said to  be a Lorentzian almost paracontact metric manifold, if it admits a (1, 1)-tensor field  ϕ, a contravariant vector field ζ, a 1-form ν and a Lorentzian metric g satisfying  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1)  ν(ζ) + 1 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2)  ϕ2E = E + ν(E)ζ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3)  ϕζ = 0,  ν(ϕE) = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4)  g(ϕE, ϕF) = g(E, F) + ν(E)ν(F),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5)  g(E, ζ) = ν(E),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='6)  ϕ(E, F) = ϕ(F, E) = g(E, ϕF)  for any vector fields E, F ∈ χ(M), where χ(M) is the Lie algebra of vector fields  on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  If ζ is a killing vector field, the (para) contact structure is called a K-(para) contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  In such a case, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='7)  ∇Eζ = ϕE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Recently, the authors Haseeb and Prasad defined and studied the following notion:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' A Lorentzian almost paracontact manifold M is called Lorentzian  para-Kenmostu manifold if [11]  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='8)  (∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE  for any E, F on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  In an LP-Kenmostu manifold, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9)  ∇Eζ = −E − ν(E)ζ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='10)  (∇Eν)F = −g(E, F) − ν(E)ν(F),  where ∇ denotes the Levi-Civita connection respecting to the Lorentzian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Furthermore, in an LP-Kenmotsu manifold, the following relations hold [11]:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='11)  g(R(E, F)G, ζ) = ν(R(E, F)G) = g(F, G)ν(E) − g(E, G)ν(F),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='12)  R(ζ, E)F = −R(E, ζ)F = g(E, F)ζ − ν(F)E,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='13)  R(E, F)ζ = ν(F)E − ν(E)F,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='14)  R(ζ, E)ζ = E + ν(E)ζ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15)  S(E, ζ) = (n − 1)ν(E), S(ζ, ζ) = −(n − 1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='16)  Qζ = (n − 1)ζ  for any E, F, G ∈ χ(M), where R, S and Q represent the curvature tensor, the Ricci  tensor and the Q Ricci operator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS  3  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' [21] An LP-Kenmotsu manifold M is said to be ν-Einstein man-  ifold if its S(̸= 0) is of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='17)  S(E, F) = ag(E, F) + bν(E)ν(F),  where a and b are smooth functions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' In particular, if b = 0, then M is termed  as an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' [12] In an LP-Kenmotsu manifold of n-dimension, S is of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='18)  S(E, F) = (  r  n − 1 − 1)g(E, F) + (  r  n − 1 − n)ν(E)ν(F),  where r is the scalar curvature of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' In an n-dimensional LP-Kenmotsu manifold, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='19)  ζ(r) = 2(r − n(n − 1)),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='20)  (∇EQ)ζ = QE − (n − 1)E,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='21)  (∇ζQ)E = 2QE − 2(n − 1)E  for any E on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='18) yields  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='22)  QE = (  r  n − 1 − 1)E + (  r  n − 1 − n)ν(E)ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Taking the covariant derivative of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='22) with respect to F and making use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9)  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='10), we lead to  (∇F Q)E = F(r)  n − 1(E + ν(E)ζ) − (  r  n − 1 − n)(g(E, F)ζ + ν(E)F + 2ν(E)ν(F)ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  By contracting F in the foregoing equation and using trace {F → (∇F Q)E} =  1  2E(r), we find  n − 3  2(n − 1)E(r) =  � ζ(r)  n − 1 − (r − n(n − 1))  �  ν(E),  which by replacing E by ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' We refer the readers to see  [14] for the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' From the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='19), it is noticed that if an n-dimensional  LP-Kenmotsu manifold possesses the constant scalar curvature, then r = n(n − 1)  and hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='18) reduces to S(E, F) = (n − 1)g(E, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, the manifold under  consideration is an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Ricci-Yamabe solitons on LP-Kenmotsu manifolds  Let the metric of an n-dimensional LP-Kenmotsu manifold be a Ricci-Yamabe  soliton (g, K, Λ, σ, ρ), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' By differentiating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) covariantly with  resprct to G, we have  (∇G£Kg)(E, F)  =  −2σ(∇GS)(E, F) + ρ(Gr)g(E, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1)  Since ∇g = 0, then the following formula [20]  (£K∇Eg −∇E£Kg −∇[K,E]g)(F, G) = −g((£K∇)(E, F), G)−g((£K∇)(E, G), F)  4  MOBIN AHMAD, GAZALA AND MOHD BILAL  turns to  (∇E£Kg)(F, G) = g((£K∇)(E, F), G) + g((£K∇)(E, G), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Since the operator £K∇ is symmetric, therefore we have  2g((£K∇)(E, F), G) = (∇E£Kg)(F, G) + (∇F £Kg)(E, G) − (∇G£Kg)(E, F),  which by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) takes the form  2g((£K∇)(E, F), G)  =  −2σ[(∇ES)(F, G) + (∇F S)(G, E) + (∇GS)(E, F)]  +ρ[(Er)g(F, G) + (Fr)g(G, E) + (Gr)g(E, F)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2)  Putting F = ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5), we find  2g((£K∇)(E, ζ), G)  =  −2σ[(∇ES)(ζ, G) + (∇ζS)(G, E) − (∇GS)(E, ζ)]  +ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3)  By virtue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3) leads to  2g((£K∇)(E, ζ), G)  =  −4σ[S(E, G) − (n − 1)g(E, G)]  +ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  By eliminating G from the foregoing equation, we have  2(£K∇)(F, ζ)  =  ρg(Dr, F)ζ − ρ(Dr)ν(F) − 4σQF  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4)  +[4σ(n − 1) + 2ρ(r − n(n − 1))]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  If we take r as constant, then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='19) we find r = n(n − 1), and hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4)  reduces to  (£K∇)(F, ζ)  =  −2σQF + 2σ(n − 1)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5)  Taking covariant derivative of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5) with respect to E, we have  (∇E£K∇)(F, ζ)  =  (£K∇)(F, E) − 2σν(E)[QF − (n − 1)F]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='6)  −  2σ(∇EQ)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Again from [20], we have  (£KR)(E, F)G = (∇E£K∇)(F, G) − (∇F £K∇)(E, G),  which by putting G = ζ and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='6) takes the form  (£KR)(E, F)ζ  =  2σν(F)(QE − (n − 1)E) − 2σν(E)(QF − (n − 1)F)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='7)  −2σ((∇EQ)F − (∇F Q)E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Putting F = ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='7) then using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='21), we arrive at  (£KR)(E, ζ)ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='8)  The Lie derivative of R(E, ζ)ζ = −E − ν(E)ζ along K leads to  (£KR)(E, ζ)ζ − g(E, £Kζ)ζ + 2ν(£Kζ)E = −(£Kν)(E)ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9), we have  (£Kν)(E)ζ = −2ν(£Kζ)E + g(E, £Kζ)ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='10)  Taking the Lie derivative of g(E, ζ) = ν(E), we find  (£Kν)(E) = g(E, £Kζ) + (£Kg)(E, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='11)  By putting F = ζ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15), we have  (£Kg)(E, ζ) = −{2σ(n − 1) + 2Λ − ρn(n − 1)}ν(E),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='12)  A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS  5  where r = n(n − 1) being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  The Lie derivative of g(ζ, ζ) = −1 along K we lead to  (£Kg)(ζ, ζ) = −2ν(£Kζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='13)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='16), we find  ν(£Kζ) = −{σ(n − 1) + Λ − ρn(n − 1)  2  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='14)  Now, combining the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='17), we find  Λ = ρn(n − 1)  2  − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15)  Thus, we have  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Let (M, g) be an n-dimensional LP-Kenmotsu manifold admitting  Ricci-Yamabe soliton (g, K, Λ, σ, ρ) with constant scalar curvature tensor, then Λ =  ρn(n−1)  2  − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  For σ = 1 and ρ = 0, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15) we have Λ = −(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, we have the  following:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If an n-dimensional LP-Kenmotsu manifold admits a Ricci soliton  with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  For σ = 0 and ρ = 1, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15) we have Λ =  n(n−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, we have the  following:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If an n-dimensional LP-Kenmotsu manifold admits a Yamabe  soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  For σ = 1 and ρ = −1, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15) we have Λ = − (n2−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, we have the  following:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If an n-dimensional LP-Kenmotsu manifold admits an Einstein  soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Now, we consider the metric of an n-dimensional LP-Kenmotsu manifold as a  Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ), then from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9) we have  S(E, F) = − 1  σ (Λ − 1 − ρr  2 )g(E, F) + 1  σ ν(E)ν(F),  where σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='16)  By putting F = ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='16) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15), we find  Λ = ρr  2 − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='17)  Now, comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='17), we have r = n−1  σ  + n(n − 1), which by using in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='17) it follows that Λ = −σ(n − 1) + ρ(n−1)(1+nσ)  2σ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, we have the following  theorem:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' An n-dimensional LP-Kenmotsu manifold with constant scalar  curvature admitting Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ) is an ν-Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Moreover, the soliton is expanding, steady or shrinking according to ρ  σ > 2σ − ρn,  ρ  σ = 2σ − ρn, or ρ  σ < 2σ − ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  6  MOBIN AHMAD, GAZALA AND MOHD BILAL  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Gradient Ricci-Yamabe solitons on LP-Kenmotsu manifolds  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' A Riemannian (or semi-Riemannian) metric g on M is called a  gradient RYS, if  Hessv + σS + (Λ − ρr  2 )g = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1)  where Hessv denotes the Hessian of a smooth function v on M and defined by  Hessv = ∇∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Let M be an n-dimensional LP-Kenmotsu manifold with g as a gradient RYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Then equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1) can be written as  ∇EDv + σQE + (Λ − ρr  2 )E = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2)  for all vector fields E on M, where D denotes the gradient operator of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Taking  the covariant derivative of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2) with respect to F, we have  ∇F ∇EDv = −σ{(∇F Q)E + Q(∇F E)} + ρF(r)  2  E − (Λ − ρr  2 )∇F E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3)  Interchanging E and F in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3), we lead to  ∇E∇F Dv = −σ{(∇EQ)F + Q(∇EF)} + ρE(r)  2  F − (Λ − ρr  2 )∇EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4)  By making use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4), we find  R(E, F)Dv = σ{(∇F Q)E − (∇EQ)F} + ρ  2{E(r)F − F(r)E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5)  Now, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='18), we find  QE = (  r  n − 1 − 1)E + (  r  n − 1 − n)ν(E)ζ,  which on taking covariant derivative with repect to F leads to  (∇F Q)E  =  F(r)  n − 1(E + ν(E)ζ) − (  r  n − 1 − n)(g(E, F)ζ  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='6)  +2ν(E)ν(F)ζ + ν(E)F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  By using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='6) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5), we have  R(E, F)Dv  =  (n − 1)ρ − 2σ  2(n − 1)  {E(r)F − F(r)E} +  σ  n − 1{F(r)ν(E)ζ − E(r)ν(F)ζ}  −σ(  r  n − 1 − n)(ν(E)F − ν(F)E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='7)  Contracting forgoing equation along E gives  S(F, Dv)  =  �(n − 1)2ρ − 2σ(n − 2)  n − 1  �  F(r)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='8)  +σ(n − 3)(r − n(n − 1))  n − 1  ν(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  From the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='18), we can write  S(F, Dv) = (  r  n − 1 − 1)F(v) + (  r  n − 1 − n)ν(F)ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9)  A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS  7  Now, by equating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='9), then putting F = ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='19), we  find  ζ(v) = r − n(n − 1)  n − 1  {2(n − 1)ρ − σ(5n − 13)  n − 1  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='10)  Taking the inner product of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='7) with ζ, we get  F(v)ν(E) − E(v)ν(F) = ρ  2{E(r)ν(F) − F(r)ν(E)},  which by replacing E by ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='10), we infer  F(v) = −(r − n(n − 1)){3ρ − σ(5n − 13)  (n − 1)2 }ν(F) − ρ  2F(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='11)  If we take r as constant, then from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5, we get r = n(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='11)  leads to F(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' This implies that v is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, the soliton under the  consideration is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Hence we state:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If the metric of an LP-Kenmotsu manifold of constant scalar curva-  ture tensor admitting a special type of vector field is gradient RYS, then the soliton  is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  For v constant, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2) turns to  σQE = −(Λ − ρr  2 )E,  which leads to  S(E, F) = − 1  σ (Λ − ρn(n − 1)  2  )g(E, F),  σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='12)  By putting E = F = ζ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='12) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='15), we obtain  Λ = ρn(n − 1)  2  − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='13)  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If an n-dimensional LP-Kenmotsu manifold admits a gradient  Ricci soliton with the constant scalar curvature, then the manifold under the con-  sideration is an Einstein manifold and Λ = ρn(n−1)  2  − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  For σ = 1 and ρ = 0, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='13) we find Λ = −(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thu, we have the  following:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If an n-dimensional LP-Kenmotsu manifold admits a gradient  Ricci soliton with the constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  For σ = 1 and ρ = −1, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='13) we have Λ = − (n−1)(n+2)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, we have  the following:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' If an n-dimensional LP-Kenmotsu manifold admits an gradient  Einstein soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' We consider the 5-dimensional manifold M 5 =  �  (x1, x2, x3, x4, x5) ∈ R5 : x5 > 0  �  ,  where (x1, x2, x3, x4, x5) are the standard coordinates in R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Let ̺1, ̺2, ̺3, ̺4 and  ̺5 be the vector fields on M 5 given by  ̺1 = ex5 ∂  ∂x1  , ̺2 = ex5 ∂  ∂x2  , ̺3 = ex5 ∂  ∂x3  , ̺4 = ex5 ∂  ∂x4  , ̺5 =  ∂  ∂x5  = ζ,  8  MOBIN AHMAD, GAZALA AND MOHD BILAL  which are linearly independent at each point of M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Let g be the Lorentzian metric  defined by  g(̺i, ̺i) = 1,  for  1 ≤ i ≤ 4  and  g(̺5, ̺5) = −1,  g(̺i, ̺j) = 0,  for  i ̸= j,  1 ≤ i, j ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Let ν be the 1-form defined by ν(E) = g(E, ̺5) = g(̺, ζ) for all E ∈ χ(M 5), and  let ϕ be the (1, 1)-tensor field defined by  ϕ̺1 = −̺2, ϕ̺2 = −̺1, ϕ̺3 = −̺4, ϕ̺4 = −̺3, ϕ̺5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  By applying linearity of ϕ and g, we have  ν(ζ) = g(ζ, ζ) = −1, ϕ2E = E + ν(E)ζ and g(ϕE, ϕF) = g(E, F) + ν(E)ν(F)  for all E, F ∈ χ(M 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus for ̺5 = ζ, the structure (ϕ, ζ, ν, g) defines a Lorentzian  almost paracontact metric structure on M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Then we have  [̺i, ̺j] = −̺i,  for  1 ≤ i ≤ 4, j = 5,  [̺i, ̺j] = 0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  By using Koszul’s formula, we can easily find we obtain  ∇̺i̺j =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  −̺5,  1 ≤ i = j ≤ 4,  −̺i,  1 ≤ i ≤ 4, j = 5,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Also one can easily verify that  ∇Eζ = −E − η(E)ζ  and  (∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Therefore, the manifold is an LP-Kenmotsu manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  From the above results, we can easily obtain the non-vanishing components of R  as follows:  R(̺1, ̺2)̺1 = −̺2, R(̺1, ̺2)̺2 = ̺1, R(̺1, ̺3)̺1 = −̺3, R(̺1, ̺3)̺3 = ̺1,  R(̺1, ̺4)̺1 = −v4, R(̺1, ̺4)̺4 = ̺1, R(̺1, ̺5)̺1 = −̺5, R(̺1, ̺5)̺5 = −̺1,  R(̺2, ̺3)̺2 = −̺3, R(̺2, ̺3)̺3 = ̺2, R(̺2, ̺4)̺2 = −̺4, R(̺2, ̺4)̺4 = ̺2,  R(̺2, ̺5)̺2 = −̺5, R(̺2, ̺5)̺5 = −̺2, R(̺3, ̺4)̺3 = −̺4, R(̺3, ̺4)̺4 = ̺3,  R(̺3, ̺5)̺3 = −̺5, R(̺3, ̺5)̺5 = −̺3, R(̺4, ̺5)̺4 = −̺5, R(̺4, ̺5)̺5 = −̺4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Also, we calculate the Ricci tensors as follows:  S(̺1, ̺1) = S(̺2, ̺2) = S(̺3, ̺3) = S(̺4, ̺4) = 4,  S(̺5, ̺5) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Therefore, we have  r = S(̺1, ̺1) + S(̺2, ̺2) + S(̺3, ̺3) + S(̺4, ̺4) − S(̺5, ̺5) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Now by taking Dv = (̺1v)̺1 + (̺2v)̺2 + (̺3v)̺3 + (̺4v)̺4 + (̺5v)̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' we have  ∇̺1Dv = (̺1(̺1v) − (̺5v))̺1 + (̺1(̺2v))̺2 + (̺1(̺3v))̺3 + (̺1(̺4v))̺4 + (̺1(̺5v) − (̺1v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ∇̺2Dv = (̺2(̺1v))̺1 + (̺2(̺2v) − (̺5v))̺2 + (̺2(̺3v))̺3 + (̺2(̺4v))̺4 + (̺2(̺5v) − (̺2v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ∇̺3Dv = (̺3(̺1v))̺1 + (̺3(̺2v))̺2 + (̺3(̺3v) − (̺5v))̺3 + (̺3(̺4v))̺4 + (̺3(̺5v) − (̺3v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ∇̺4Dv = (̺4(̺1v))̺1 + (̺4(̺2v))̺2 + (̺4(̺3v))̺3 + (̺4(̺4v) − (̺5v))̺4 + (̺4(̺5v) − (̺4v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ∇̺5Dv = (̺5(̺1v))̺1 + (̺5(̺2v))̺2 + (̺5(̺3v))̺3 + (̺5(̺4v))̺4 + (̺5(̺5v))̺5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS  9  Thus, by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' we obtain  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ̺1(̺1v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺2(̺2v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺3(̺3v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺4(̺4v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺5(̺5v) = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺1(̺2v) = ̺1(̺3v) = ̺1(̺4v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺2(̺1v) = ̺2(̺3v) = ̺2(̺4v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺3(̺1v) = ̺3(̺2v) = ̺3(̺4v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺4(̺1v) = ̺4(̺2v) = ̺4(̺3v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺1(̺5v) − (̺1v) = ̺2(̺5v) − (̺2v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  ̺3(̺5v) − (̺3v) = ̺4(̺5v) − (̺4v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='14)  Thus, the equations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='14) are respectively amounting to  e2x5 ∂2v  ∂x2  1  − ∂v  ∂x5  = −(Λ + 4σ − 10ρ),  e2x5 ∂2v  ∂x2  2  − ∂v  ∂x5  = −(Λ + 4σ − 10ρ),  e2x5 ∂2v  ∂x2  3  − ∂v  ∂x5  = −(Λ + 4σ − 10ρ),  e2x5 ∂2v  ∂x2  4  − ∂v  ∂x5  = −(Λ + 4σ − 10ρ),  ∂2v  ∂x2  5  = −(Λ + 4σ − 10ρ),  ∂2v  ∂x1∂x2  =  ∂2v  ∂x1∂x3  =  ∂2v  ∂x1∂x4  =  ∂2v  ∂x2∂x3  =  ∂2v  ∂x2∂x4  =  ∂2v  ∂x3∂x4  = 0,  ex5  ∂2v  ∂x5∂x1  + ∂v  ∂x1  = ex5  ∂2v  ���x5∂x2  + ∂v  ∂x2  = ex5  ∂2v  ∂x5∂x3  + ∂v  ∂x3  = ex5  ∂2v  ∂x5∂x4  + ∂v  ∂x4  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  From the above equations it is observed that v is constant for Λ = −4σ + 10ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Hence, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Thus, g is a gradient RYS with the soliton vector  field K = Dv, where v is constant and Λ = −4σ + 10ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Hence, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2 is  verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='  10  MOBIN AHMAD, GAZALA AND MOHD BILAL  [5] Chidananda,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='4134/CKMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=', Ricci-Yamabe maps for Riemannian flows and their volume  variation and volume entropy, Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=', The Ricci Flow on Surfaces, Mathematics and General Relativity (Santa  Cruz, CA, 1986), Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=' and De, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='  [10] Haseeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' and Almusawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='  [11] Haseeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' and Prasad, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=', Certain results on Lorentzian para-Kenmotsu manifolds, Bol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Parana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='  [12] Haseeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=', Some results on Lorentzian para-Kenmotsu manifolds, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Transilvania Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='1155/2022/5718736  [14] Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=' and Harry, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='geomphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='104547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  [16] Pankaj, Chaubey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content=', On Kenmotsu manifolds admitting η-Ricci-Yamabe solitons,Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='  [20] Yano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
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+page_content='  I, Marcel Dekker, New York, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  [21] Yano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' and Kon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=', Structures on manifolds, World Scientific, (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Mobin Ahmad  Department of Mathematics,  Integral University, Kursi Road,  Lucknow-226026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Email : mobinahmad68@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='com  Gazala  Department of Mathematics,  Integral University, Kursi Road,  Lucknow-226026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Email : gazala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='com  Mohd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content=' Bilal  Department of Mathematical Sciences,  Umm Ul Qura University,  Makkah, Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='  Email: mohd7bilal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'}

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+arXiv:2301.01498v1  [math.RT]  4 Jan 2023
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+Abstract. g-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined
+by tilting theory.
+We give a classification of complete g-fans of rank 2.
+More explicitly, our
+first main result asserts that every complete sign-coherent fan of rank 2 is a g-fan of some
+finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem,
+Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level
+of finite dimensional algebras. Our second main result gives a necessary and sufficient condition
+for algebras of rank 2 to be g-convex.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries
+5
+2.1.
+Preliminaries on fans
+5
+2.2.
+Sign-coherent fans of rank 2
+6
+3.
+Basic results in silting theory
+10
+3.1.
+Preliminaries
+10
+3.2.
+Silting complexes in terms of matrices
+12
+3.3.
+Uniserial property of g-finite algebras
+14
+4.
+Gluing, Rotation and Subdivision of g-fans
+15
+4.1.
+Gluing fans
+15
+4.2.
+Rotation and Mutation
+17
+4.3.
+Subdivision and Extension
+19
+4.4.
+Proof of Theorem 1.3
+22
+4.5.
+Gluing fans II
+23
+5.
+g-Convex algebras of rank 2
+26
+5.1.
+Characterizations of g-convex algebras of rank 2
+26
+5.2.
+Proof of Theorem 5.1
+27
+Acknowledgments
+29
+References
+29
+1. Introduction
+The notion of tilting complexes is central to control equivalences of derived categories. The
+class of silting complexes [KV] gives a completion of the class of tilting complexes with respect to
+mutation, which is an operation to replace a direct summand of a given silting complex to construct
+a new silting complex [AI]. The subclass of 2-term silting complexes enjoys remarkable properties
+[AIR, DF]. They give rise to a fan in the real Grothendieck group of a finite dimensional algebra
+A, see e.g. [H1, H2, Pl, B, DIJ, BST, As].
+In our previous article [AHIKM1], we introduced a g-fan Σ(A) of A and established a basic
+theory of g-fans and the associated g-polytopes. A g-fan of each finite dimensional algebra A
+belongs to the following special class of nonsingular fans [AHIKM1, Proposition 4.12].
+Definition 1.1. A sign-coherent fan is a pair (Σ, σ+) satisfying the following conditions.
+1
+
+2
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+(a) Σ is a nonsingular fan in Rd.
+(b) σ+, −σ+ ∈ Σd.
+(c) Take a Z-basis e1, . . . , ed of Zd such that σ+ = cone{ei | 1 ≤ i ≤ d}, and denote the orthant
+corresponding to ǫ ∈ {±1}d by
+Rd
+ǫ := cone{ǫ(1)e1, . . . , ǫ(d)ed} = {x1e1 + · · · + xded | ǫ(i)xi ≥ 0 for each 1 ≤ i ≤ d}.
+Then for each σ ∈ Σ, there exists ǫ ∈ {±1}d such that σ ⊆ Rd
+ǫ.
+We denote by Fansc(d) the set of complete sign-coherent fans in Rd, and by k-Fan(d) the set of
+complete g-fans of finite dimensional k-algebras of rank d. Note that a g-fan Σ(A) is complete if
+and only if A is g-finite (Proposition 3.9). Then we have
+Fansc(d) ⊃ k-Fan(d).
+It is very natural to study the following problem.
+Problem 1.2. Characterize complete sign-coherent fans in Rd which can be realized as a g-fan of
+some finite dimensional algebra.
+This paper is devoted to give a complete answer to this problem for the case d = 2. The result
+was very simple and came as a surprise to us.
+Theorem 1.3 (Theorem 4.13). For each field k, we have
+Fansc(2) = k-Fan(2).
+Thus any complete sign-coherent fan in R2 can be realized as a g-fan of some finite dimensional
+k-algebra.
+We explain our method to prove Theorem 1.3. Each sign-coherent fan of rank 2 is obtained by
+gluing two fans of the following form.
+Σ =
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+Σ′ =
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+Recall that a finite dimensional k-algbera Λ is elementary if the k-algebra Λ/JΛ is isomorphic to
+a product of k. This is automatic if Λ is basic and k is algebraically closed. We prove Gluing
+Theorem 4.1, which asserts that if both Σ and Σ′ are g-fans of finite dimensional elementary k-
+algebras, then so is their gluing. Therefore by symmetry, it suffices to consider sign-coherent fans
+Σ of the form above. Now such Σ can be obtained from the fan
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+by applying subdivision in the fourth quadrant repeatedly. We prove Rotation Theorem 4.3 and
+Subdivision Theorem 4.7, which imply that if Σ is a g-fan of a finite dimensional k-algebra, then
+so are the subdivisions of Σ in the fourth quadrant.
+Figure 1 gives fans in Fan+−
+sc (2) with at most 8 facet, where each edge shows a subdivision.
+Figure 2 gives examples of algebras whose g-fans are given in Figure 1.
+For each finite dimensional algebra A, we define a g-polytope P(A) by gluing each simplex
+associated with the cones in Σ(A).
+If P(A) is convex, we call Σ(A) convex and A g-convex.
+For example, Brauer tree algebras A are g-convex, and this fact plays an important role in the
+classification of 2-term tilting complexes of A [AMN]. From tilting theoretic point of view, g-convex
+algebras are the most fundamental. Therefore it is important to study the following problem.
+Problem 1.4. Classify convex g-fans in Rd.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+3
+Σ00
+Σ111
+Σ2121
+Σ1212
+Σ31221
+Σ22131
+Σ12213
+Σ21312
+Σ13122
+Σ412221
+Σ321321
+Σ313131
+Σ312312
+Σ231231
+Σ222141
+Σ221412
+Σ122214
+Σ123123
+Σ214122
+Σ131313
+Σ213213
+Σ132132
+Σ141222
+Figure 1. Fans in Fan+−
+sc (2) with at most 8 facets
+An answer to the case d = 2 was given in [AHIKM1, Theorem 6.3]. There are precisely 7 convex
+g-fans up to isomorphism of g-fans.
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+•+
+−
+❄❄❄❄❄
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+•+
+−
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+•+
+−
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❖❖❖❖❖❖❖
+❄❄❄❄❄
+•+
+−
+❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄
+❖❖❖❖❖❖❖
+❄❄❄❄❄
+•+
+−
+❄❄❄❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖❖❖❖❖❖❖
+❄❄❄❄❄
+•+
+−
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄
+✴✴✴✴✴✴✴
+❄❄❄❄❄
+❖❖❖❖❖❖❖
+❄❄❄❄❄
+More precisely, in the last Section 5, we show that there are 16 convex g-fans in Fansc(2)
+Σa;b with a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.
+We also give a characterization of algebras whose g-fans are one of them. Let t(ΛM) (respectively,
+t(MΛ)) be the minimal number of generators of a left (respectively, right) Λ-module M.
+
+4
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+Σ00
+k [ •
+• ]
+Σ111
+k
+�
+•
+•
+a �
+�
+Σ2121
+k
+�
+•
+•
+a �
+c
+�
+�
+⟨c2⟩
+Σ1212
+k
+�
+•
+•
+a �
+c
+�
+�
+⟨c2⟩
+Σ31221
+k
+
+
+•
+•
+a �
+c1
+�
+c2
+�
+
+
+⟨c2
+1, c2
+2, c2c1, ac2⟩
+Σ22131
+k
+
+
+•
+•
+a �
+b
+�
+c0
+�
+c1
+�
+
+
+�b2, c2
+0, c2
+1, c1c0,
+ba − ac0
+�
+Σ12213
+k
+
+
+•
+•
+a �
+c1 �
+c2
+�
+
+
+�
+c2
+1, c2
+2, c1c2, c2a
+�
+Σ21312
+k
+�
+•
+•
+a �
+b
+�
+c
+�
+�
+⟨b2, c2, bac⟩
+Σ13122
+k
+
+
+•
+•
+a �
+c0 �
+c1
+�
+b
+�
+
+
+�b2, c2
+0, c2
+1, c0c1,
+ab − c0a
+�
+Σ412221
+k
+
+
+•
+•
+a �
+c1
+�
+c2
+�
+c3
+�
+
+
+�c2
+1, c2
+2, c2
+3, c3c2, c3c1,
+c2c1, c1c3, ac2, ac3
+�
+Σ321321
+k
+
+
+•
+•
+a �
+b
+�
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+�
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+
+
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+1, c2
+2, ac0,
+ac2, c2c1, c2c0, c0c2,
+c0c1c0, ba − ac1c0
+�
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+k
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+2, ac2,
+c2c1, c2c0, c1c0, c0c2,
+c0c1c2, ba − ac0
+�
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+k
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+a �
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+bac1, c2c1
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+c1
+�
+c2
+�
+
+
+�b2
+0, b2
+1, c2
+0, c2
+1, c2
+2, ac2, b1b0,
+c2c1c2, c2c0, c1c0, c0c2,
+b0a − ac0, b1a − ac1c2
+�
+Σ222141
+k
+
+
+•
+•
+a �
+b0 �
+b1
+�
+c0
+�
+c1
+�
+c2
+�
+
+
+�b2
+0, b2
+1, c2
+0, c2
+1, c2
+2, b1b0,
+c2c1, c2c0, c1c0, c0c2
+b0a − ac0, b1a − ac1
+�
+Σ221412
+k
+
+
+•
+•
+a �
+b0 �
+b1
+�
+c0
+�
+c1
+�
+
+
+�b2
+0, b2
+1, c2
+0, c2
+1, b1b0,
+b0b1, c1c0, b1ac0,
+b1ac1, b0a − ac0
+�
+Σ122214
+k
+
+
+•
+•
+a �
+c1 �
+c2
+�
+c3
+�
+
+
+�c2
+1, c2
+2, c2
+3, c2c3, c1c3
+c1c2, c3c1, c2a, c3a
+�
+Σ123123
+k
+
+
+•
+•
+a �
+c0 �
+c1
+�
+c2
+�
+b
+�
+
+
+� b2, c2
+0, c2
+1, c2
+2, c0a,
+c2a, c1c2, c0c2, c2c0,
+c0c1c0, ab − c0c1a
+�
+Σ214122
+k
+
+
+•
+•
+a �
+c0 �
+c1
+�
+b0
+�
+b1
+�
+
+
+�b2
+0, b2
+1, c2
+0, c2
+1, b0b1,
+b1b0, c0c1, c0ab1,
+c1ab1, ab0 − c0a
+�
+Σ131313
+k
+
+
+•
+•
+a �
+c0 �
+c1
+�
+c2
+�
+b
+�
+
+
+�
+b2, c2
+0, c2
+1, c2
+2, c2a,
+c1c2, c0c2, c0c1, c2c0,
+c2c1c0, ab − c0a
+�
+Σ213213
+k
+
+
+•
+•
+a �
+c1 �
+c2
+�
+b
+�
+
+
+�b2, c2
+1, c2
+2, c2a,
+c1ab, c1c2
+�
+Σ132132
+k
+
+���
+•
+•
+a �
+c0 �
+c1
+�
+c2
+�
+b0
+�
+b1
+�
+
+
+�b2
+0, b2
+1, c2
+0, c2
+1, c2
+2, c2a, b0b1,
+c2c1c2, c0c2, c0c1, c2c0,
+ab0 − c0a, ab1 − c2c1a
+�
+Σ141222
+k
+
+
+•
+•
+a �
+c0 �
+c1
+�
+c2
+�
+b0
+�
+b1
+�
+
+
+�b2
+0, b2
+1, c2
+0, c2
+1, c2
+2, b0b1,
+c2c0, c1c2, c0c2, c0c1,
+ab0 − c0a, ab1 − c1a
+�
+Figure 2. Algebras whose g-fans are given in Figure 1
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+5
+Theorem 1.5 (Theorem 5.1). Let A be a basic finite dimensional algebra, {e1, e2} a complete set
+of primitive orthogonal idempotents in A, and Pi = eiA (i = 1, 2).
+(a) A is g-convex if and only if Σ(A) = Σa;b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.
+(b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)). Then we have the following statements.
+• Σ(A) = Σ00;b for some b if and only if (l, r) = (0, 0).
+• Σ(A) = Σ111;b for some b if and only if (l, r) = (1, 1).
+• Σ(A) = Σ1212;b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some
+left generator x of e1Ae2 and Rx := {a ∈ e1Ae1 | ax ∈ xAe2}.
+• Σ(A) = Σ2121;b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some
+right generator x of e1Ae2 and Lx := {b ∈ e2Ae2 | xb ∈ e1Ax}.
+Σ00;b
+•
+P2
+P1
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+⑧⑧⑧⑧⑧⑧
+Σ111;b
+•
+P2
+P1
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+Σ1212;b
+•
+P2
+P1
+❄
+❄
+❄
+❄
+❄
+❄
+✴✴✴✴✴✴✴✴✴
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+Σ2121;b
+•
+P2
+P1
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+Further, in a forthcoming paper [AHIKM2], we will give a complete answer to Problem 1.4 for
+d = 3.
+2. Preliminaries
+2.1. Preliminaries on fans. We recall some fundamental materials on fans. We refer the reader
+to e.g. [F, BR, BP] for these materials.
+A convex polyhedral cone σ is a set of the form σ = {�s
+i=1 rivi | ri ≥ 0}, where v1, . . . , vs ∈ Rd.
+We denote it by σ = cone{v1, . . . , vs}. Note that {0} is regarded as a convex polyhedral cone. We
+collect some notions concerning convex polyhedral cones. Let σ be a convex polyhedral cone.
+• The dimension of σ is the dimension of the linear space generated by σ.
+• We say that σ is strongly convex if σ ∩ (−σ) = {0} holds, i.e., σ does not contain a linear
+subspace of positive dimension.
+• We call σ rational if each vi can be taken from Qd.
+• We denote by ⟨·, ·⟩ the usual inner product.
+A supporting hyperplane of σ is a hyperplane
+{v ∈ σ | ⟨u, v⟩ = 0} in Rd given by some u ∈ Rd satisfying σ ⊂ {v ∈ Rd | ⟨u, v⟩ ≥ 0}.
+• A face τ of σ is the intersection of σ with a supporting hyperplane of σ.
+In what follows, a cone means a strongly convex rational polyhedral cone for short.
+Definition 2.1. A fan Σ in Rd is a collection of cones in Rd such that
+(a) each face of a cone in Σ is also contained in Σ, and
+(b) the intersection of two cones in Σ is a face of each of those two cones.
+For each i ≥ 0, we denote by Σi the subset of cones of dimension i. For example, Σ0 consists of
+the trivial cone {0}. We call each element in Σ1 a ray of Σ.
+We collect some notions concerning fans used in this paper. Let Σ be a fan in Rd.
+• We call Σ finite if it consists of a finite number of cones.
+• We call Σ complete if �
+σ∈Σ σ = Rd.
+• We call Σ nonsingular (or smooth) if each maximal cone in Σ is generated by a Z-basis for Zd.
+We prepare some notions which will be used in this paper.
+Definition 2.2. Let Σ be a nonsingular fan in Rd. We call Σ pairwise positive if the following
+condition is satisfied.
+
+6
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+• For each two adjacent maximal cones σ, τ ∈ Σd, take Z-basis {v1, . . . , vd−1, vd} and {v1, . . . , vd−1, v′
+d}
+of Zd such that σ = cone{v1, . . . , vd−1, vd} and τ = cone{v1, . . . , vd−1, v′
+d}. Then vd + v′
+d belongs
+to cone{v1, . . . , vd−1}.
+Definition 2.3. Let Σ and Σ′ be fans in Rd and Rd′ respectively.
+(1) An isomorphism Σ ≃ Σ′ of fans is an isomorphism Zd ≃ Zd′ of abelian groups such that
+the induced linear isomorphism Rd → Rd′ gives a bijection Σ ≃ Σ′ between cones.
+(2) Let (Σ, σ+) and (Σ′, σ′
++) be sign-coherent fans in Rd and Rd′ respectively. An isomorphism
+of sign-coherent fans is an isomorphism f : Σ ≃ Σ′ of fans such that {f(σ+), f(−σ+)} =
+{σ′
++, −σ′
++}.
+2.2. Sign-coherent fans of rank 2. In this subsection, we introduce some terminologies of sign-
+coherent fans of rank 2, and discuss some fundamental properties.
+Let Σ be a complete nonsingular fan of rank 2. We denote the rays of Σ by
+v1, v2, . . . , vn−1, vn = v0
+(2.1)
+which are indexed in a clockwise orientation. For each 1 ≤ i ≤ n, since Σ is nonsingular, there
+exists an integer ai satisfying
+aivi = vi−1 + vi+1 for each 1 ≤ i ≤ n.
+We call the sequence of integers
+s(Σ) = (a1, . . . , an)
+(2.2)
+the defining sequence of Σ. In fact, Σ is uniquely determined by its defining sequence. A fan with
+defining sequence (a1, . . . , an) is denoted by
+Σ(a1, . . . , an).
+Remark 2.4. [F, Section 2.5] An integer sequence (a1, . . . , an) is a defining sequence of nonsingular
+complete fan of rank 2 if and only if it satisfies
+n
+�
+i=1
+ai = 3n − 12 and
+�0
+−1
+1
+a1
+� �0
+−1
+1
+a2
+�
+· · ·
+�0
+−1
+1
+an
+�
+=
+�1
+0
+0
+1
+�
+.
+Definition 2.5. We denote by Fansc(2) the set of all (possibly infinite) fans Σ satisfying that
+• Σ is a sign-coherent fans (Definition 1.1) of rank 2 with positive and negative cones
+σ+ := cone{(1, 0), (0, 1)} and σ− := cone{(−1, 0), (0, −1)} respectively,
+• each ray is a face of precisely two facets.
+We denote the subset of complete fans by
+Fansc(2) ⊂ Fansc(2).
+For Σ ∈ Fansc(2), we denote the rays of Σ in a clockwise orientation by
+Σ1 = {v1 := (1, 0), v2, . . . , vn−1, vn = v0 := (0, 1)}.
+Then there exists 2 ≤ i ≤ n − 2 such that vi = (0, −1) and vi+1 = (−1, 0).
+•
+vn=v0=(0,1)
+v1=(1,0)
+vi=(0,−1)
+vi+1=(−1,0)
++
+−
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+In this case, it is more convenient to rewrite (2.2) as
+s(Σ) = (a1, a2, . . . , ai; an, an−1, . . . , ai+1).
+Thus we mainly use the notation
+Σ(a1, . . . , ai; an, . . . , ai+1) = Σa1,...,ai;an,...,ai+1
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+7
+instead of Σ(a1, . . . , an).
+We consider subsets
+Fan
++−
+sc (2)
+⊂
+Fansc(2)
+∪
+∪
+Fan+−
+sc (2)
+⊂
+Fansc(2)
+which consist of fans Σ containing σ−+ := cone{(−1, 0), (0, 1)}, i.e. Σ has the following form.
+•
+σ−+
++
+−
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+Thus the rays and the facets of Σ ∈ Fan+−
+sc (2) are written as
+Σ1
+=
+{v1 = (1, 0), v2, . . . , vn−2 = (0, −1), vn−1 = (−1, 0), vn = v0 = (0, 1)},
+(2.3)
+Σ2
+=
+{σ1, . . . , σn−3, σn−2 = σ−, σn−1 = σ−+, σn = σ+}.
+(2.4)
+Similarly, we define Fan
+−+
+sc (2) and Fan−+
+sc (2) as the subsets of Fansc(2) and Fansc(2) respectively
+which consist of fans containing σ+− := cone{(1, 0), (0, −1)}.
+The following observations are clear.
+Lemma 2.6. The following assertions hold.
+(1) The correspondence Σ �→ {−σ | σ ∈ Σ} gives bijections Fan
++−
+sc (2) → Fan
+−+
+sc (2) and Fan+−
+sc (2) →
+Fan−+
+sc (2).
+(2) Let Σ ∈ Fansc(2). Then Σ ∈ Fan+−
+sc (2) (respectively, Σ ∈ Fan−+
+sc (2)) holds if and only if s(Σ)
+has the form
+(b1, . . . , bm; 0, 0) (respectively, (0, 0; b1, . . . , bm)).
+In this case, bi ≥ 0 holds for any 1 ≤ i ≤ m.
+Definition 2.7. For Σ ∈ Fan
++−
+sc (2) and Σ′ ∈ Fan
+−+
+sc (2), we define Σ ∗ Σ′ ∈ Fansc(2) by
+(Σ ∪ Σ′)1
+:=
+Σ1 ∪ Σ′
+1
+(Σ ∪ Σ′)2
+:=
+(Σ2 \ {σ−+}) ∪ (Σ′
+2 \ {σ+−}) .
+Σ =
+•
++
+−
+?
+σ−+
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+Σ′ =
+•
++
+−
+!
+σ+−
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+Σ ∗ Σ′ =
+•
++
+−
+?
+!
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+Then, we clearly have
+Fansc(2)
+=
+Fan
++−
+sc (2) ∗ Fan
+−+
+sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan
++−
+sc (2), Σ′ ∈ Fan
+−+
+sc (2)},
+Fansc(2)
+=
+Fan+−
+sc (2) ∗ Fan−+
+sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan+−
+sc (2), Σ′ ∈ Fan−+
+sc (2)}.
+(2.5)
+Definition 2.8. Let Σ be a (possibly infinite) nonsingular fan of rank 2. For a cone σ := cone{u, v}
+of Σ, we define a new nonsingular fan Dσ(Σ) by
+Dσ(Σ)1
+=
+Σ1 ∪ {cone{u + v}},
+Dσ(Σ)2
+=
+(Σ2 \ {σ}) ⊔ {cone{u, u + v}, cone{v, u + v}}.
+We call Dσ(Σ) the subdivision of Σ at σ.
+Σ =
+•
+..........................................
+σ
+❨
+❨
+❨
+❨
+❨
+❨
+❨
+❡
+❡
+❡
+❡
+❡
+❡
+❡
+Dσ(Σ) =
+•
+.......................................... ❨
+❨
+❨
+❨
+❨
+❨
+❨
+❡
+❡
+❡
+❡
+❡
+❡
+❡
+❡❡❡❡❡❡❡
+❨❨❨❨❨❨❨
+
+8
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+For a sequence a = (a1 . . . , an) and 1 ≤ i ≤ n, we define a new sequence by
+Di(a) = (a1, . . . , ai−1, ai + 1, 1, ai+1 + 1, ai+2, . . . , an).
+(2.6)
+For a complete nonsingular fan Σ with rays (2.1) and σi := cone{vi, vi+1} for 1 ≤ i ≤ n, we have
+s ◦ Dσi(Σ) = Di ◦ s(Σ).
+(2.7)
+Example 2.9. Figure 1 gives fans in Fan+−
+sc (2) with at most 8 facets, where
+Σa1,...,an := Σ(a1, . . . , an; 0, 0)
+and each edge shows a subdivision. Figure 2 gives examples of algebras whose g-fans are given
+in Figure 1. For example, Σ111 is the g-vector fan of a cluster algebra of type A2 [FZ1, FZ2].
+Similarly, Σ1212 and Σ2121 are the g-vector fans of cluster algebras of type B2, and Σ131313 and
+Σ313131 are the g-vector fans of cluster algebras of type G2.
+Later we need the following observation (cf. [F, Section 4.3]).
+Proposition 2.10. Each fan in Fan+−
+sc (2) can be obtained from Σ(0, 0; 0, 0) by a sequence of
+subdivisions.
+Σ(0, 0; 0, 0) =
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+To prove this, we need the following preparation.
+Lemma 2.11 (cf. [F, p.43]). Let Σ ∈ Fan+−
+sc (2) and s(Σ) = (a1, . . . , an−2; 0, 0). If n ≥ 5, then
+there exists 2 ≤ i ≤ n − 3 satisfying ai = 1.
+Proof. Let vi = (xi, yi) ∈ Z2 for 1 ≤ i ≤ n. Assume that n ≥ 5 and ai ≥ 2 for any 2 ≤ i ≤ n − 3.
+We claim that xi+1 ≥ xi holds for each 1 ≤ i ≤ n − 3. In fact, n ≥ 5 implies x2 ≥ 1 = x1. Then
+we have
+xi+1 = aixi − xi−1 ≥ 2xi − xi−1 ≥ xi
+for each 2 ≤ i ≤ n − 3, and the claim follows inductively. Consequently 1 = x1 ≤ x2 ≤ · · · ≤
+xn−2 = 0 holds, a contradiction.
+□
+We are ready to prove Proposition 2.10.
+Proof of Proposition 2.10. Let F ⊂ Fan+−
+sc (2) be the set of fans obtained from Σ(0, 0; 0, 0) by a
+sequence of subdivisions. It suffices to show Fan+−
+sc (2) = F.
+We will show that each Σ ∈ Fan+−
+sc (2) belongs to F by using induction on n = #Σ2.
+Clearly n ≥ 4 holds. If n = 4, then Σ = Σ(0, 0; 0, 0) ∈ F.
+Suppose that Σ with #Σ2 = n ≥ 5 belongs to Fan+−
+sc (2). In terms of (2.3) and (2.4), there
+exists 2 ≤ i ≤ n − 3 satisfying vi = vi−1 + vi+1 by Lemma 2.11. Since vi−1, vi+1 forms a Z-basis
+of Z2, we obtain a new fan Σ′ ∈ Fan+−
+sc (2) by
+Σ′
+1
+:=
+Σ1 \ {vi},
+Σ′
+2
+:=
+(Σ2 \ {σi−1, σi}) ∪ {σ} for σ := cone{vi−1, vi+1}.
+Since #Σ′
+2 = n − 1, the induction hypothesis implies Σ′ ∈ F. Thus Σ = Dσ(Σ′) ∈ F holds.
+□
+Remark 2.12. For each n ≥ 1, we have a bijection
+{Σ ∈ Fan+−
+sc (2) | #Σ2 = n + 3} ≃ {the ways to parenthesize n factors completely},
+where parentheses show how cones in the fourth quadrant are obtained by iterated subdivisions.
+For example, Σ141222 in Figure 1 has 5 cones σ1, . . . , σ5 in the fourth quadrant in terms of (2.4),
+and they are parenthesized as σ1(((σ2σ3)σ4)σ5). In particular, we have
+#{Σ ∈ Fan+−
+sc (2) | #Σ2 = n + 3} = 1
+n
+�2n − 2
+n − 1
+�
+.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+9
+We also have a bijection
+{Σ ∈ Fan+−
+sc (2) | #Σ2 = n + 3} ≃ {Triangulations of a regular (n + 1)-gon},
+where Σa1,...,an+1 corresponds to a triangulation satisfying the following condition: Let 1, 2, . . ., n+
+1 be the vertices of the regular (n + 1)-gon in a clockwise direction, and ai (1 ≤ i ≤ n + 1) the
+number of triangles containing the vertex i in the triangulation. For example, Σ141222 corresponds
+to the following triangulation, where 1 is the top vertex.
+We introduce piecewise linear transformation of sign coherent fan of rank 2. This is a general-
+ization of mutation of g-vectors of cluster algebras of rank 2 [FZ2, NZ], and also a special case of
+so called combinatorial mutation [ACGK, FH].
+Definition 2.13. For Σ ∈ Fan
++−
+sc (2) with σ+ = cone{(0, 1), (1, 0)}, take σ = cone{(1, 0), (ℓ, −1)} ∈
+Σ2. Define a new sign-coherent fan Σ′ by
+Σ′
+1
+:=
+(Σ1 \ {(0, 1)}) ∪ {(−ℓ, 1)}
+Σ′
+2
+:=
+(Σ2 \ {σ+, σ−+}) ∪ {−σ, cone{(−ℓ, 1), (1, 0)}},
+where the positive and negative cones of Σ′ are σ and −σ respectively.
+Σ =
+•
+(0,1)
+(1,0)
+(ℓ,−1)
+(0,−1)
+(−1,0)
++
+−
+σ
+σ−+
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+ρ(Σ) ≃ Σ′ =
+•
+(−ℓ,1)
+(1,0)
+(ℓ,−1)
+(0,−1)
+(−1,0)
++
+−
+σ
+−σ
+❄❄❄❄❄❄
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+We define the rotation ρ(Σ) ∈ Fan
++−
+sc (2) of Σ as the image of Σ′ by a linear transformation of R2
+mapping (1, 0) �→ (0, 1) and (ℓ, −1) �→ (1, 0).
+We give basic properties of rotation, where the name “rotation” is explained by (a) below.
+Proposition 2.14. Let Σ ∈ Fan+−
+sc (2) with facets (2.4) and s(Σ) = (a1, . . . , an−2; 0, 0).
+(a) We have
+s(ρ(Σ)) = (a2, . . . , an−2, a1; 0, 0).
+In particular, ρn−2(Σ) = Σ holds, and therefore ρ is an invertible operation.
+(b) For each 1 ≤ i ≤ n − 3, we have
+Dσi(Σ) = ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ).
+Proof. (a) Recall Σ1 = {v1, . . . , vn} and aivi = vi−1 + vi+1 for 1 ≤ i ≤ n. Moreover
+ρ(Σ)1 = {w1, . . . , wn} where wi := vi+1 (i ̸= n − 1), wn−1 := −v2.
+Hence we have
+wi−1 + wi+1
+=
+vi + vi+2 = ai+1vi+1 = ai+1wi for i ̸= n − 2, n,
+wn−1 + w1
+=
+−v2 + v2 = 0 · wn,
+wn−3 + wn−1
+=
+vn−2 − v2 = −(vn + v2) = −a1v1 = a1vn−1 = a1wn−2.
+Thus s(ρ(Σ)) = (a2, . . . , an−2, a1; 0, 0) as desired.
+(b) By (a), we have s ◦ ρi+1(Σ) = (ai+2, . . . , an−2, a1, . . . , ai+1; 0, 0). Thus
+s ◦ Dσn−3 ◦ ρi+1(Σ)
+(2.7)
+= Dn−3 ◦ s ◦ ρi+1(Σ) = (ai+2, . . . , an−2, a1, . . . , ai−1, ai + 1, 1, ai+1 + 1; 0, 0).
+
+10
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+By (a) again, we have
+s ◦ ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ)
+=
+(a1, . . . , ai−1, ai + 1, 1, ai+1 + 1, ai+2, . . . , an−2; 0, 0)
+=
+Di ◦ s(Σ)
+(2.7)
+= s ◦ Dσi(Σ).
+Since a fan is uniquely determined by its defining sequence, the assertion follows.
+□
+3. Basic results in silting theory
+3.1. Preliminaries. Let A be a finite dimensional algebra over a field k. Let K0(proj A) be the
+Grothendieck group of the additive category proj A, which is identified with the Grothendieck
+group of the triangulated category Kb(proj A).
+We recall basic results on silting theory from
+[AI, AIR, AHIKM1]. First we recall the definition of 2-term silting complexes.
+Definition 3.1. Let T = (T i, di) ∈ Kb(proj A).
+(a) T is called presilting if HomKb(proj A)(T, T [ℓ]) = 0 for all positive integers ℓ.
+(b) T is called silting if it is presilting and Kb(proj A) = thick T .
+(c) T is called 2-term if T i = 0 for all i ̸= 0, −1. In this case, the class [T ] = [T 0] − [T −1] ∈
+K0(proj A) of T is called the g-vector of T .
+(d) An element of K0(proj A) is rigid if it is a g-vector of some 2-term presilting complex.
+We denote by siltA (respectively, psiltA, 2-siltA, 2-psiltA) the set of isomorphism classes of basic
+silting (respectively, presilting, 2-term silting, 2-term presilting) complexes of Kb(proj A). Note
+that a 2-term presilting complex T is silting if and only if |T | = |A| holds.
+For T, U ∈ siltA, we write T ≥ U if HomKb(proj A)(T, U[ℓ]) = 0 holds for all positive integers ℓ.
+Then (siltA, ≥) is a partially ordered set [AI].
+In this paper, the subposet (2-siltA, ≥) of (siltA, ≥) plays a central role.
+It is known that
+Hasse(2-siltA) is n-regular for n := |A|. More precisely, let T = T1 ⊕ · · · ⊕ Tn ∈ 2-siltA with
+indecomposable Ti. For each 1 ≤ i ≤ n, there exists precisely one T ′ ∈ 2-siltA such that T ′ =
+T ′
+i ⊕ (�
+j̸=i Tj) for some T ′
+i ̸= Ti. In this case, we call T ′ mutation of T at Ti and write
+T ′ = µTi(T ) = µi(T ).
+In this case, either T > T ′ or T ′ < T holds. We denote T ′ by µ−
+i (T ) (respectively, µ+
+i (T )) if
+T > T ′ and call it left mutation (respectively, right mutation). The following result is fundamental
+in silting theory.
+Proposition 3.2. Let T, T ′ ∈ 2-siltA. Take a decomposition T = T1⊕· · ·⊕Tn with indecomposable
+Ti. Then the following conditions are equivalent.
+(a) T > T ′, and T and T ′ are mutation of each other.
+(b) There is an arrow T → T ′ in Hasse(2-siltA).
+(c) T ′ = T ′
+i ⊕ (�
+j̸=i Tj) and there is a triangle
+Ti
+f−→ Ui → T ′
+i → Ti[1]
+such that f is a minimal left (add �
+j̸=i Tj)-approximation.
+(d) T ′ = T ′
+i ⊕ (�
+j̸=i Tj) and there is a triangle
+Ti → Ui
+g−→ T ′
+i → Ti[1]
+such that g is a minimal right (add �
+j̸=i Tj)-approximation.
+The triangles in (c) and (d) are isomorphic, and called an exchange triangle.
+To introduce the g-fan of a finite dimensional k-algebra A, we consider the real Grothendieck
+group of A:
+K0(proj A)R := K0(proj A) ⊗Z R ≃ R|A|.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+11
+Definition 3.3. For T = T1 ⊕ · · · ⊕ Tℓ ∈ 2-psiltA with indecomposable Ti, let
+C(T )
+:=
+{
+ℓ
+�
+i=1
+ai[Ti] | a1, . . . , aℓ ≥ 0} ⊂ K0(proj A)R,
+C≤1(T )
+:=
+{
+ℓ
+�
+i=1
+ai[Ti] | a1, . . . , aℓ ≥ 0,
+ℓ
+�
+i=1
+ai ≤ 1} ⊂ K0(proj A)R.
+We call the set
+Σ(A) := {C(T ) | T ∈ 2-psiltA}
+of cones the g-fan of A. We also define the g-polytope P(A) of A by
+P(A) :=
+�
+T ∈2-siltA
+C≤1(T ).
+We say that A is g-convex if the g-polytope P(A) is convex.
+Notice that Σ(A) can be an infinite set. We give the following basic properties of g-fans.
+Proposition 3.4. Let A be a finite dimensional algebra over a field k and n := |A|.
+(a) Σ is a pairwise positive sign-coherent fan whose positive (respectively, negative) cone is given
+by σ+ := C(A) (respectively, σ− := C(A[1])).
+(b) Any cone in Σ(A) is a face of a cone of dimension n.
+(c) Any cone in Σ(A) of dimension n − 1 is a face of precisely two cones of dimension n.
+The following basic observation will be used frequently.
+Proposition 3.5. Let Λ be a finite dimensional algebra with orthogonal primitive idempotents
+1 = e1 + e2. Under the identification P1 = (1, 0) and P2 = (0, 1), the following assertions hold.
+(a) cone{(−1, 0), (0, 1)} ∈ Σ(Λ) if and only if e2Λe1 = 0.
+(b) cone{(1, 0), (0, −1)} ∈ Σ(Λ) if and only if e1Λe2 = 0.
+Proposition 3.5 is explained by the following picture.
+e2Λe1 = 0 ⇔
+•
++
+−
+P1
+P2
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+e2Λe1 = 0 ⇔
+•
++
+−
+P1
+P2
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+Proof. We only prove (a): Σ(A) ∈ Fan+−
+sc (2) if and only if P1[1] ⊕ P2 ∈ 2-siltA if and only if
+HomKb(proj A)(P1, P2) = 0 if and only if e2Λe1 = 0.
+□
+We end this subsection with recalling the sign decomposition technique studied in [Ao, AHIKM1].
+We have to introduce the following notations.
+Definition 3.6. Let A be a basic finite dimensional algebra over a field k with |A| = n, and
+1 = e1 + · · · + en the orthogonal primitive idempotents. For ǫ ∈ {±1}n, we define
+K0(proj A)ǫ,R := cone(ǫi[eiA] | i ∈ {1, . . ., n})
+and a subfan of Σ(A) by
+Σǫ(A) := {σ ∈ Σ(A) | σ ⊂ K0(proj A)ǫ,R}.
+Define idempotents of A by
+e+
+ǫ :=
+�
+ǫi=1
+ei and e−
+ǫ :=
+�
+ǫi=−1
+ei.
+We denote by Aǫ the subalgebra of A given by
+Aǫ :=
+� e+
+ǫ Ae+
+ǫ
+e+
+ǫ Ae−
+ǫ
+0
+e−
+ǫ Ae−
+ǫ
+�
+.
+
+12
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+Define an ideal Iǫ of Aǫ by
+Iǫ :=
+� rad(e+
+ǫ Ae+
+ǫ ) ∩ Anne+
+ǫ Ae+
+ǫ (e+
+ǫ Ae−
+ǫ )
+0
+0
+rad(e−
+ǫ Ae−
+ǫ ) ∩ Ann(e+
+ǫ Ae−
+ǫ )e−
+ǫ Ae−
+ǫ
+�
+.
+The following result is often very useful to calculate Σǫ(A).
+Proposition 3.7. [AHIKM1, Example 4.26] For each ideal I of Aǫ contained in Iǫ, the isomor-
+phisms − ⊗Aǫ A : K0(proj Aǫ)R ≃ K0(proj A)R and − ⊗Aǫ (Aǫ/I) : K0(proj Aǫ)R ≃ K0(proj Aǫ/I)R
+gives an isomorphism of fans
+Σǫ(A) ≃ Σǫ(Aǫ/I).
+The following finiteness condition plays a central role in this paper.
+Definition 3.8. Let A be a finite dimensional algebra over a field k. We say that A is g-finite if
+#2-siltA < ∞. (This is called τ-tilting finite in [DIJ].)
+Proposition 3.9. A is g-finite (or equivalently, Σ(A) is finite) if and only if Σ(A) is complete.
+3.2. Silting complexes in terms of matrices. In this subsection, we give basic properties of
+2-term presilting complexes. Throughout this subsection, we assume the following.
+Assumption 3.10. For rings A and B and an Aop ⊗k B-module X which is finitely generated on
+both sides, let
+Λ :=
+� A
+X
+0
+B
+�
+.
+Equivalently, Λ is a ring with orthogonal idempotents 1 = e1 + e2 satisfying e2Λe1 = 0. In fact,
+we can recover Λ from A := e1Λe1, B := e2Λe2 and X := e1Λe2 by the equality above.
+Consider projective Λ-modules
+P1 := [A X], P2 := [0 B] ∈ proj Λ.
+For s, t ≥ 0, we denote by Ms,t(X) the set of s × t matrices with entries in X. Then we have an
+isomorphism
+Ms,t(X) ≃ HomΛ(P ⊕t
+2 , P ⊕s
+1 )
+sending x ∈ Ms,t(X) to the left multiplication x(·) : P ⊕t
+2
+→ P ⊕s
+1 . Thus we have a 2-term complex
+Px := (P ⊕t
+2
+x(·)
+−−→ P ⊕s
+1 ) ∈ per Λ.
+The following observation is basic.
+Proposition 3.11. Let s, t, u, v ≥ 0, x ∈ Ms,t(X) and y ∈ Mu,v(X).
+(a) Then we have an exact sequence
+Mu,s(A) ⊕ Mv,t(B)
+[(·)x y(·)]
+−−−−−−→ Mu,t(X) → Homper Λ(Px, Py[1]) → 0.
+(b) In particular, Px is presilting if and only if Ms,t(X) = Ms(A)x + xMt(B) holds.
+Proof. The assertion (a) follows from an exact sequence
+HomΛ(P ⊕s
+1
+, P ⊕u
+1
+) ⊕ HomΛ(P ⊕t
+2 , P ⊕v
+2
+)
+[(·)x y(·)]
+−−−−−−→ HomΛ(P ⊕t
+2 , P ⊕u
+1
+) → Homper Λ(Px, Py[1]) → 0.
+The assertion (b) is immediate from (a).
+□
+The following construction of silting complexes of Λ will be used frequently, where t(XB) (re-
+spectively, t(AX)) is the minimal number of generators of X as a right B-module (respectively,
+left A-module).
+Proposition 3.12. In Assumption 3.10, assume that A and B are local k-algebras.
+(a) Σ(Λ) contains cone{(0, −1), (1, −r)} for r := t(XB) = dim(X/XJB)B/JB.
+(b) Σ(Λ) contains cone{(1, 0), (ℓ, −1)} for ℓ := t(AX) = dimA/JA(X/JAX).
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+13
+(c) Let g1, . . . , gr be a minimal set of generators of the B-module X. Then µ+
+1 (Λ[1]) = Pg ⊕P2[1] ∈
+2-siltΛ holds for g := [g1 · · · gr] ∈ M1,r(X).
+(d) Let h1, . . . , hℓ a minimal set of generators of the Aop-module X. Then µ−
+2 (Λ) = P1 ⊕ Ph ∈
+2-siltΛ holds for h :=
+� h1
+...
+hℓ
+�
+∈ Mℓ,1(X).
+By Propositions 3.5 and 3.12, a part of Σ(Λ) has the following form.
+Σ(Λ) =
+•
+P2
+P1
+Ph=ℓP1−P2
+Pg=P1−rP2
+P2[1]
+P1[1]
++
+−
+µ−
+2 (Λ)
+µ+
+1 (Λ[1])
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+✯✯✯✯✯✯✯✯✯✯✯✯✯
+✴✴✴✴✴✴✴✴✴
+Proof. We only prove (a)(c) since (b)(d) are the duals. A minimal right (add P2[1])-approximation
+of P1[1] is given by
+g(·) : P2[1]⊕r → P1[1].
+Thus the mutation of Λ[1] at P1[1] is Pg ⊕ P2.
+□
+Now we assume that B is a local algebra. We fix a minimal set of generators g1, . . . , gr of the
+right B-module X and set
+g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB),
+where (·) is a canonical surjection X ։ X/XJB. Then we have an isomorphism
+g(·) : Mr,1(B/JB) ≃ X/XJB,
+and we define a map π : X → Mr,1(B/JB) by
+π := (X
+(·)
+−→ X/XJB
+(g(·))−1
+−−−−−→ Mr,1(B/JB)).
+For each s, t ≥ 0, an entry-wise application of π gives a map
+π : Ms,t(X) → Ms,t(Mr,1(B/JB)) = Mrs,t(B/JB).
+In other words, for the identity matrix Is ∈ Ms(k) and gIs :=
+� g
+O
+...
+O
+g
+�
+∈ Ms(M1,r(k)) =
+Ms,rs(k), we have
+x = (gIs)π(x) for each x ∈ Ms,t(X).
+(3.1)
+Define a morphism of k-algebras
+φ : Ms(A) → Mrs(B/JB) by a(gIs) = (gIs)φ(a).
+Later we will use the following observation.
+Proposition 3.13. In Assumption 3.10, assume that B is a local algebra. Let s, t ≥ 0.
+(a) π : Ms,t(X) → Mrs,t(B/JB) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard
+Mrs,t(B/JB) as an Ms(A)op-module via φ.
+(b) Let x ∈ Ms,t(X). If Px is presilting, then π(x) ∈ Mrs,t(B/JB) has full rank.
+Proof. (a) For any a ∈ Ms(A), x ∈ Ms,t(X) and b ∈ Mt(B), we need to show π(axb) = φ(a)π(x)b.
+In fact,
+(gIs)φ(a)π(x)b = a(gIs)π(x)b
+(3.1)
+= axb = axb
+(3.1)
+= (gIs)π(axb)
+gives the desired equality since gIs(·) is injective.
+
+14
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+(b) By Proposition 3.11(b), we have Ms,t(X) = Ms(A)x + xMt(B). Applying π, we have
+Mrs,t(B/JB) = π(Ms(A)x + xMt(B))
+(a)
+=
+φ(Ms(A))π(x) + π(x)Mt(B)
+⊂
+Mrs(B/JB)π(x) + π(x)Mt(B/JB).
+Thus the right-hand side is Mrs,t(B/JB). This clearly implies that π(x) has full rank.
+□
+For completeness, we also give the dual statement of Proposition 3.13. Now we assume that A
+is a local algebra. We fix a minimal set of generators h1, . . . , hℓ of the left A-module X and set
+h :=
+� h1
+...
+hℓ
+�
+∈ Mℓ,1(X) and h :=
+� h1
+...
+hℓ
+�
+∈ Mℓ,1(X/JAX),
+where by abuse of notations, (·) is a canonical surjection X ։ X/JAX. Then we have an isomor-
+phism (·)h : M1,ℓ(A/JA) ≃ X/JAX. By abuse of notations, let
+π := (X
+(·)
+−→ X/JAX
+((·)h)−1
+−−−−−→ M1,ℓ(A/JA)).
+For each s, t ≥ 0, an entry-wise application of π gives a map
+π : Ms,t(X) → Ms,t(M1,ℓ(A/JA)) = Ms,ℓt(A/JA).
+Define a morphism of k-algebras
+φ : Mt(B) → Mℓt(A/JA) by (hIt)b = φ(b)(hIs).
+We have the following dual of Proposition 3.13.
+Proposition 3.14. In Assumption 3.10, assume that A is a local algebra. Let s, t ≥ 0.
+(a) π : Ms,t(X) → Ms,ℓt(A/JA) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard
+Ms,ℓt(A/JA) as an Mt(B)-module via φ.
+(b) Let x ∈ Ms,t(X). If Px is presilting, then π(x) ∈ Ms,ℓt(A/JA) has full rank.
+3.3. Uniserial property of g-finite algebras. As an application of results in the previous sub-
+section, we prove the following result, which is not used in the rest of this paper.
+Theorem 3.15. Let Λ be a finite dimensional elementary k-algebra, and 1 = e1 + · · · + en the
+orthogonal primitive idempotents. If Λ is g-finite, then for each 1 ≤ i ̸= j ≤ n, eiΛej/eiΛejJΛej
+is a uniserial (eiΛei)op-module and eiΛej/eiJΛeiJΛej is a uniserial ejΛej-module.
+Thanks to sign decomposition, we can deduce Theorem 3.15 from the following result.
+Theorem 3.16. Let A and B be local k-algebras with k ≃ A/JA ≃ B/JB. If X is a Aop ⊗k B-
+module such that
+�
+A
+X
+0
+B
+�
+is g-finite, then X/XJB is a uniserial Aop-module and X/JAX is a
+uniserial B-module.
+Proof of Theorem 3.16⇒Theorem 3.15. Since Λ is g-finite, so is Γ := (ei + ej)Λ(ei + ej).
+By
+Proposition 3.7, Γ+− =
+� eiΛei
+eiΛej
+0
+ejΛej
+�
+is also g-finite. Thus the assertion follows from Theorem
+3.16.
+□
+In the rest of this subsection, we prove Theorem 3.16.
+The following observation plays a key role in the proof, where we identify K0(proj Λ) with Z2
+via [A X] �→ (1, 0), [0 B] �→ (0, 1).
+Lemma 3.17. Let Λ :=
+� A
+X
+0
+k
+�
+. Assume that (1, −1) ∈ K0(proj Λ) is rigid.
+(a) There exists h ∈ X such that X = Ah.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+15
+(b) Let Λ′ :=
+� A
+JAX
+0
+k
+�
+and t ≥ 1. If (1, −t) ∈ K0(proj Λ) is rigid, then (1, 1 − t) ∈ K0(proj Λ′)
+is rigid.
+Proof. (a) By Proposition 3.11(b), there exists h ∈ X satisfying X = Ah + hk = Ah.
+(b) By Proposition 3.11(b), there exists [x1 x2 · · · xt] ∈ M1,t(X) such that
+M1,t(X) = A[x1 · · · xt] + [x1 · · · xt]Mt(k).
+(3.2)
+As in Section 3.2, the element h gives surjections
+π := (X
+(·)
+−→ X/JAX
+((·)h)−1
+−−−−−→ A/JA = k) and π : M1,t(X) → M1,t(k).
+By Proposition 3.14, π(x) ∈ M1,t(k) has full rank. By changing indices if necessary, we can assume
+x1 ∈ A×h. Multiplying an element in A× from left, we can assume x1 = h. Multiplying an element
+in GLt(k) from right, we can assume xi ∈ JAh for each 2 ≤ i ≤ t. We claim
+M1,t−1(JAX) = A[x2 · · · xt] + [x2 · · · xt]Mt−1(k).
+In fact, fix any [y2 · · · yt] ∈ M1,t−1(JAX). By (3.2) there exist a ∈ A and b = [bij]1≤i,j≤t ∈ Mt(k)
+such that
+[0 y2 · · · yt] = a[h x2 · · · xt] + [h x2 · · · xt]b.
+(3.3)
+Applying π, we obtain
+[0 0 · · · 0] = a[1 0 · · · 0] + [1 0 · · · 0]b in M1,t(k).
+Thus we obtain b12 = · · · = b1t = 0. Looking at the i-th entries for 2 ≤ i ≤ t of (3.3), we have
+[y2 · · · yt] = a[x2 · · · xt] + [x2 · · · xt][bij]2≤i,j≤n.
+Thus the claim follows.
+□
+We are ready to prove Theorem 3.16.
+Proof of Theorem 3.16. We prove that X/XJB is a uniserial Aop-module under a weaker assump-
+tion that (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1. Since Λ :=
+� A
+X/XJB
+0
+k
+�
+is a factor
+algebra of Λ, the element (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1. Replacing Λ by Λ, we can
+assume that
+B = k and Λ =
+� A
+X
+0
+k
+�
+.
+We use induction on dimk X.
+By Lemma 3.17(a), the Aop-module X has a unique maximal
+submodule JAX. Let Λ′ =
+� A
+JAX
+0
+k
+�
+. By Lemma 3.17(b), (1, −t) ∈ K0(proj Λ′) is rigid for
+each t ≥ 1.
+By induction hypothesis, JAX is a uniserial Aop-module.
+Therefore X is also a
+uniserial Aop-module.
+□
+4. Gluing, Rotation and Subdivision of g-fans
+4.1. Gluing fans. Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive
+idempotents 1 = e1 + e2 ∈ Λ and 1 = e′
+1 + e′
+2 ∈ Λ′. In this subsection, we prove the following
+Gluing Theorem, where we identify K0(proj Λ) and K0(proj Λ′) with Z2 by e1Λ = (1, 0) = e′
+1Λ′ and
+e2Λ = (0, 1) = e′
+2Λ′.
+Theorem 4.1 (Gluing Theorem). Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal
+primitive idempotents 1 = e1 + e2 ∈ Λ and 1 = e′
+1 + e′
+2 ∈ Λ′. Assume e1Λe2 = 0 and e′
+2Λ′e′
+1 = 0,
+or equivalently, Σ(Λ) ∈ Fan
++−
+sc (2) and Σ(Λ′) ∈ Fan
+−+
+sc (2) (Proposition 3.5). Then, there exists an
+elementary k-algebra Γ such that
+Σ(Γ) = Σ(Λ) ∗ Σ(Λ′).
+(4.1)
+
+16
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+Theorem 4.1 is explained by the following picture.
+Σ(Λ) =
+•
++
+−
+?
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+Σ(Λ′) =
+•
++
+−
+!
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+Σ(Γ) =
+•
++
+−
+?
+!
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+The construction of Γ is as follows: We can write
+Λ =
+�
+A
+X
+0
+B
+�
+and Λ′ =
+�
+C
+0
+Y
+D
+�
+,
+where A, B, C, D are local k-algebras, X is an Aop ⊗k B-module, and Y is an Dop ⊗k C-module.
+Since Λ and Λ′ are elementary, we have k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD. Let A ×k C be a
+fiber product of canonical surjections (·) : A → k and (·) : C → k, that is,
+A ×k C := {(a, c) ∈ A × C | a = c}.
+Let B ×k D be a fibre product of (·) : B → k and (·) : D → k. Using the projections A ×k C → A
+and B ×k D → B, we regard X as an (A ×k C)op ⊗k (B ×k D)-module, and using the projections
+A ×k C → C and B ×k D → D, we regard Y as an (B ×k D)op ⊗k (A ×k C)-module.
+We prove that the algebra
+Γ :=
+� A ×k C
+X
+Y
+B ×k D
+�
+satisfies Σ(Γ) = Σ(Λ) ∗ Σ(Λ′), where the multiplication of the elements of X and those of Y are
+defined to be zero.
+Proof of Theorem 4.1. It suffices to prove
+Σ+−(Γ) = Σ+−(Λ) and Σ−+(Γ) = Σ−+(Λ′).
+For ǫ = (+, −), we have Γǫ =
+�
+A ×k C
+X
+0
+B ×k D
+�
+. The ideal I :=
+�
+rad C
+0
+0
+rad D
+�
+of Γǫ is
+contained in Iǫ, and we have an isomorphism Γǫ/I ≃ Λ of k-algebras. Applying Proposition 3.7 to
+Γ, we get Σ+−(Γ) = Σ+−(Λ). By the same argument, Σ−+(Γ) = Σ−+(Λ′) holds, as desired.
+□
+Example 4.2. Let Λ and Λ′ be the following algebras.
+Λ :=
+k
+
+
+1
+2
+a3 �
+a4
+�
+a2
+�
+a1
+�
+
+
+⟨a2
+1, a2
+2, a2
+4, a2a1, a2a3 − a3a4⟩,
+Λ′ :=
+k
+�
+1
+2
+b1
+�
+b2
+�
+�
+⟨b2
+2⟩
+By Examples 4.6 and 4.11 below, we have
+Σ(Λ) = Σ13122;00 =
+Σ(Λ′) = Σ00;1212 =
+Let A = e1Λe1, X = e1Λe2, B = e2Λe2, C = e1Λ′e1, Y = e2Λ′e1, D = e2Λ′e2 and Γ =
+� A ×k C
+X
+Y
+B ×k D
+�
+. Then
+Γ =
+k
+
+
+1
+2
+a3 �
+a4
+�
+a2
+�
+a1
+� b1
+�
+b2
+�
+
+
+⟨a2
+1, a2
+2, a2
+4, a2a1, a2a3 − a3a4, b2
+2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+17
+By Gluing Theorem 4.1, we have
+Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) = Σ13122;1212 =
+4.2. Rotation and Mutation. In this subsection, we explain a connection between the rotation
+of a fan given in Definition 2.13 and mutation of a 2-term silting complex.
+The following main result in this section shows that mutation of an algebra induce the rotation
+of the g-fan, where we identify K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1).
+Theorem 4.3 (Rotation Theorem). Let Λ be a finite dimensional k-algebra of rank 2 with or-
+thogonal primitive idempotents 1 = e1 + e2. Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan
++−
+sc (2)
+(Proposition 3.5). Then, there exists a finite dimensional k-algebra Γ such that
+Σ(Γ) = ρ(Σ(Λ)).
+Furthermore, if Λ is elementary, then Γ can be taken to be elementary.
+Theorem 4.3 is explained by the following picture.
+Σ(Λ) =
+•
++
+−
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+Σ(Γ) ≃
+•
++
+−❄❄❄❄❄❄
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+To prove Theorem 4.3, we need the following preparation.
+Let A be a basic finite dimensional algebra over a field k with |A| = n, and 1 = e1 + · · · + en
+the orthogonal primitive idempotents. For 1 ≤ i ≤ n and δ ∈ {±1}, consider a half space
+Rn
+i,δ := {x1e1 + · · · + xden ∈ Rn | δxi ≥ 0}
+and define a subfan of Σ by
+Σi,δ := {σ ∈ Σ | σ ⊂ Rn
+i,δ}.
+On the other hand, for elements T ≥ T ′ in siltA, we consider the interval
+[T ′, T ] := {U ∈ siltA | T ≥ U ≥ T ′}.
+The following result provides a correspondence of a part of two g-fans.
+Proposition 4.4. For 1 ≤ i ≤ n, let B := EndA(µ−
+i (A)), where µ−
+i (A) = Ti ⊕ (�
+j̸=i P A
+j ).
+(a) [AHIKM1, Threom 4.26] There exists a triangle functor F : Kb(proj A) → Kb(proj B) which
+satisfies F(Ti) ≃ P B
+i
+and F(P A
+j ) ≃ P B
+j
+for each j ̸= i and gives an isomorphism K0(proj A) ≃
+K0(proj B) and an isomorphism of fans
+Σi,−(A) ≃ Σi,+(B).
+(b) There are isomorphisms (1 − ei)A(1 − ei) ≃ (1 − ei)B(1 − ei) and A/(1 − ei) ≃ B/(1 − ei) of
+k-algebras.
+Proof. (b) Although this is known to experts, we give a proof for convenience of the reader. The first
+isomorphism is clear. To prove the second one, notice that A/(1 − ei) = EndKb(proj A)(P A
+i )/[A/P A
+i ]
+and B/(1−ei) = EndKb(proj A)(Ti)/[T/Ti] hold, where [X] denotes the ideal consisting of morphisms
+factoring through add X. Let Pi
+f−→ Q
+g−→ Ti
+h−→ Pi[1] be an exchange triangle. Let a ∈ eiAei =
+EndKb(proj A)(Pi). Since f is a minimal left (add A/Pi)-approximation of Pi, we obtain the following
+commutative diagram.
+Pi
+f
+�
+a�
+Q
+g
+�
+�
+Ti
+h �
+b�
+Pi[1]
+a[1]
+�
+Pi
+f
+� Q
+g
+� Ti
+h � Pi[1]
+
+18
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+It is routine to check that the desired isomorphism A/(1 − ei)A = EndKb(proj A)(P A
+i )/[A/P A
+i ] ≃
+B/(1 − ei) = EndKb(proj A)(Ti)/[T/Ti] is given by a �→ b.
+□
+We are ready to prove Theorem 4.3.
+Proof of Theorem 4.3. Let T = P Λ
+1 ⊕ T2 := µ−
+2 (Λ) and E := EndKb(proj Λ)(T ). By Proposition
+4.4(a), we have a triangle functor F : Kb(proj Λ) → Kb(proj E) which satisfies
+F(P Λ
+1 ) = P E
+1
+and F(T2) = P E
+2
+and induces an isomorphism F : K0(proj Λ) ≃ K0(proj E) and an isomorphism of fans
+F : Σ2,−(Λ) ≃ Σ2,+(E).
+•
+Σ(Λ)
+P Λ
+2
+P Λ
+1
+T2
+T
++
+−
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+•
+Σ(E)
+P E
+2 [1]
+P E
+1
+P E
+2
+P E
+1 [1]
++
+−
+E
+E[1]
+❄❄❄❄❄❄
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+•
+Σ(Γ)
+P Γ
+2 [1]
+P Γ
+1
+P Γ
+2
+P Γ
+1 [1]
++
+−
+Γ
+Γ[1]
+❄❄❄❄❄❄
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+Applying Theorem 3.7 to E, we obtain a k-algebra Γ := E−+ such that
+e1Γe2 = 0 and Σ−+(Γ) = Σ−+(E).
+Therefore under the isomorphism K0(proj Γ) ≃ Z2 given by P Γ
+1 �→ (0, 1) and P Γ
+2 �→ (1, 0), we have
+Σ(Γ) = ρ(Σ(Λ)), as desired.
+It remains to prove the last assertion. By Proposition 4.4(a), we have isomorphisms e1Ee1 ≃
+e1Λe1 and Λ/(e1) ≃ E/(e1) of k-algebras. Thus, if Λ is elementary, then so are E and Γ.
+□
+We give two examples of Theorem 4.3. The first one satisfies E = Γ.
+Example 4.5. Let Λ be the following algebra. Then Σ(Λ) is the following fan by Example 4.11
+below.
+Λ =
+k
+�
+1
+2
+a �
+b
+�
+�
+⟨b2⟩
+Σ(Λ) = Σ1212 =
+We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ
+a·
+−→ e1Λ] ⊕ e1Λ and E := EndKb(proj Λ)(T ). Then we have
+Γ = E =
+k
+�
+1
+2
+a �
+b
+�
+�
+⟨b2⟩
+and Σ(Γ) = ρ(Σ(Λ)) = Σ2121 =
+The second example satisfies E ̸= Γ.
+Example 4.6. Let Λ be the following algebra. Then Σ(Λ) is the following fan by Example 4.12
+below.
+Λ =
+k
+�
+1
+2
+a �
+b
+�
+c
+�
+�
+⟨b2, c2, bac⟩
+Σ(Λ) = Σ21312 =
+We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ ( a·
+ac·)
+−−−−→ e1Λ⊕2] ⊕ e1Λ and E := EndKb(proj Λ)(T ), where we
+switch the indices 1 and 2 unlike the proof of Theorem 4.3. Then
+E =
+k
+�
+1
+2
+a �
+a′
+�
+b
+�
+�
+⟨b2, a′b, a′aa′⟩
+and Σ(E) = Σ13122;111 =
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+19
+where new arrows a, a′ and b are morphisms in Kb(proj Λ) given by commutative diagrams
+0
+e1Λ
+e2Λ
+e1Λ⊕2
+�
+�
+( a·
+ac·) �
+( 0
+1)
+�
+e2Λ
+e1Λ⊕2
+0
+e1Λ
+( a·
+ac·) �
+�
+�
+( 0 b· )
+�
+e2Λ
+e1Λ⊕2
+e2Λ
+e1Λ⊕2
+( a·
+ac·) �
+c· �
+( a·
+ac·) �
+( 0 1
+0 0)
+�
+respectively. Let Γ := E+− =
+� e1Ee1
+e1Ee2
+0
+e2Ee2
+�
+=
+� ⟨e1, b, aa′, baa′⟩k
+⟨a, ba, aa′a, baa′a⟩k
+0
+⟨e2, a′a⟩k
+�
+.
+Then
+Γ =
+k
+
+
+1
+2
+a �
+c
+�
+b
+�
+b′
+�
+
+
+⟨b2, b′2, c2, b′b, b′a − ac⟩ and Σ(Γ) = ρ(Σ(Λ)) = Σ13122 =
+where b′ := aa′ and c := a′a.
+4.3. Subdivision and Extension. In this section, we realize subdivisions of g-fans of rank 2 by
+extensions of algebras. The following theorem is a main result of this section, where we identify
+K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1).
+Theorem 4.7 (Subdivision Theorem). Let Λ be a finite dimensional elementary k-algebra with
+orthogonal primitive idempotents 1 = e1+e2. Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan
++−
+sc (2)
+(Proposition 3.5). Then, for cones σ = C(µ+
+1 (Λ[1])) and σ′ := C(µ−
+2 (Λ)) of Σ(Λ), there exist finite
+dimensional elementary k-algebras Γ and Γ′ such that
+Σ(Γ) = Dσ(Σ(Λ)) and Σ(Γ′) = Dσ′(Σ(Λ)).
+Theorem 4.7 is explained by the following picture.
+Σ(Λ) =
+•
+P2
+P1
+P2[1]
+P1[1]
++
+−
+µ−
+2 (Λ)
+µ+
+1 (Λ[1])
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+✯✯✯✯✯✯✯✯✯✯✯✯✯
+✴✴✴✴✴✴✴✴✴
+Σ(Γ) =
+•
++
+−
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+✯✯✯✯✯✯✯✯✯✯✯✯✯
+✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬
+✯✯✯✯✯✯✯✯✯✯✯✯✯
+Σ(Γ′) =
+•
++
+−
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄❄❄❄❄❄
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+✯✯✯✯✯✯✯✯✯✯✯✯✯
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❲
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+✴✴✴✴✴✴✴✴✴
+In the rest, we only prove the existence of Γ since the existence of Γ′ is the dual.
+The construction of Γ is as follows:
+Construction 4.8. By Proposition 3.5, we can write
+Λ =
+� A
+X
+0
+B
+�
+.
+where A, B are local k-algebras and X is an Aop ⊗k B-module. Since Λ is elementary, we have
+k ≃ A/JA ≃ B/JB. Let
+X := X/XJB.
+Then the k-dual DX is an A-module, and we regard it as an Aop-module by using the action of k
+through the natural surjection A → k. Let
+C := A ⊕ DX
+be a trivial extension algebra of A by DX. Let
+(·) : A → k, (·) : B → k and (·) : X → X
+
+20
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+be canonical surjections. We regard
+Y :=
+� k
+X
+�
+as a Cop ⊗k B-module by
+(a, f) · [ α
+x ] · b :=
+�
+aαb+f(x)b
+axb
+�
+for (a, f) ∈ C = A ⊕ DX, [ α
+x ] ∈ Y =
+� k
+X
+�
+and b ∈ B.
+Then we set
+Γ :=
+�
+C
+Y
+0
+B
+�
+.
+In the rest of this subsection, we prove Theorem 4.7. We set
+Q1 := [C Y ], Q2 := [0 B] ∈ proj Γ.
+For y ∈ Ms,t(Y ) ≃ HomΓ(Q⊕t
+2 , Q⊕s
+1 ), we define
+Qy := [Q⊕t
+2
+y(·)
+−−→ Q⊕s
+1 ] ∈ Kb(proj Γ).
+We fix a minimal set of generators g1, . . . , gr of the B-module X. Then (g1, . . . , gr) forms a
+k-basis of X = X/XJB. Set
+g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB).
+We need the following easy observation.
+Lemma 4.9. Σ(Γ) contains cone{(0, 1), (1, −r−1)} and cone{(1, −r−1), (1, −r)}. More explicitly,
+let
+� 0
+g
+�
+∈ M1,r(Y ) and
+� 0 1
+g 0
+�
+∈ M1,r+1(Y ).
+Then Q� 0 1
+g 0
+� ⊕ Q2[1] and Q� 0
+g
+� ⊕ Q� 0 1
+g 0
+� belong to 2-siltΓ.
+Proof. A minimal set of generators of the B-module Y is given by the r+1 columns of
+� 0 1
+g 0
+�
+. Thus
+Q� 0 1
+g 0
+� ⊕ Q2[1] ∈ 2-siltΓ holds by Proposition 3.12.
+In the rest, we prove that T := Q� 0
+g
+� ��� Q� 0 1
+g 0
+� is basic silting. By the first statement, Q� 0 1
+g 0
+�
+is indecomposable. If Q� 0
+g
+� is not indecomposable, then |T | is bigger than two, a contradiction.
+Thus T is basic.
+We will show that T is presilting by using Proposition 3.11(b). By our choice of g, we have
+gMr,1(B) = X and (DX)g = M1,r(k).
+Thus we have
+� 0
+g
+�
+Mr(B) = M1,r([ 0
+X ]) and (DX)
+� 0
+g
+�
+= M1,r([ k
+0 ]), and hence
+C
+� 0
+g
+�
++
+� 0
+g
+�
+Mr(B) ⊃ (DX)
+� 0
+g
+�
++
+� 0
+g
+�
+Mr(B) = M1,r([ 0
+X ]) + M1,r([ k
+0 ]) = M1,r(Y ).
+This clearly implies
+C
+� 0
+g
+�
++
+� 0 1
+g 0
+�
+Mr+1,r(B) = M1,r(Y ),
+and a similar argument implies
+C
+� 0 1
+g 0
+�
++
+� 0
+g
+�
+Mr,r+1(B) = M1,r+1(Y ).
+Thus Proposition 3.11(b) implies that T is presilting, as desired.
+□
+As in Section 3.2, the element g gives a surjection
+π := (X
+(·)
+−→ X
+(g(·))−1
+−−−−−→ Mr,1(B) = Mr,1(k)),
+which extends to the map π : Ms,t(X) → Mrs,t(k) for each s, t ≥ 0.
+The following observation is crucial.
+Proposition 4.10. Let s, t ≥ 0. For x ∈ Ms,t(X), consider [ 0
+x ] ∈ Ms,t(Y ).
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+21
+(a) Px is indecomposable in Kb(proj Λ) if and only if Q[ 0
+x] is indecomposable in Kb(proj Γ).
+(b) If Q[ 0
+x] is a presilting complex of Γ, then Px is a presilting complex of Λ.
+(c) The converse of (b) holds if t ≤ rs.
+(d) The restriction of Σ(Γ) to {(x, y) ∈ R2 | 0 ≤ −y ≤ rx} coincides with that of Σ(Λ).
+Proof. Notice that Γ is the trivial extension Λ ⊕ I of Λ by the following ideal I of Γ:
+I :=
+�
+DX
+k
+0
+0
+�
+.
+(a) Since Px ≃ Q[ 0
+x] ⊗Γ Λ and Q[ 0
+x] ≃ Px ⊗Λ Γ, the assertion follows immediately.
+(b) Since Λ = Γ/I and Q[ 0
+x] ⊗Γ Λ ≃ Px, the assertion follows.
+(c) Assume that Px is a presilting complex of Λ. Then by Proposition 3.11(b), we have
+Ms,t(X) = Ms(A)x + xMt(B).
+(4.2)
+Again by Proposition 3.11(b), it suffices to show the equality
+V := Ms(C) [ 0
+x ] + [ 0
+x ] Mt(B) = Ms,t(
+� k
+X
+�
+).
+Since V ⊃ Ms(A) [ 0
+x ] + [ 0
+x ] Mt(B)
+(4.2)
+= Ms,t([ 0
+X ]) holds, it suffices to show
+V ⊃ Ms,t([ k
+0 ]).
+(4.3)
+By our assumption t ≤ rs and Proposition 3.13(b), π(x) has rank t and the map
+(·)π(x) : Ms,rs(k) → Ms,t(k)
+(4.4)
+is surjective. We denote by g∗
+1, . . . , g∗
+r the basis of DX which is dual to g1, . . . , gr. Then the map
+(·)
+� g∗
+1
+...
+g∗
+r
+�
+: M1,r(k) ≃ DX is a bijection, and we denote its inverse by
+π′ : DX ≃ M1,r(k).
+It gives a bijection π′ : Ms(DX) ≃ Ms,rs(k). We have a commutative diagram
+Ms(DX) × Ms,t(X)
+π′×π
+�
+eval.
+�❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+❱
+Ms,rs(k) × Mrs,t(k)
+mult.
+�
+Ms,t(k)
+where eval. is given by the evaluation map DX × X → DX × X → k. Thus the commutativity of
+the diagram above and the surjectivity of (4.4) shows that the map
+(·)x : Ms(DX) → Ms,t(k)
+is also surjective. Therefore the desired claim (4.3) follows from
+V ⊃ Ms(C) [ 0
+x ] ⊃ Ms(DX) [ 0
+x ] = Ms,t([ k
+0 ]).
+□
+(d) Immediate from (c).
+We are ready to prove Theorem 4.7.
+Proof of Theorem 4.7. The assertion follows from Lemma 4.9 and Proposition 4.10(d).
+□
+We give two examples of Subdivision Theorem 4.7.
+
+22
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+Example 4.11. Let Λ be the following algebra. Then Σ(Λ) is the following fan.
+Λ = k[1 → 2]
+Σ(Λ) = Σ111 =
+Applying Theorem 4.7 to Λ, we get
+Γ :=
+�
+k ⊕ Dk
+� k
+k
+�
+0
+k
+�
+=
+k
+�
+1
+2
+�
+b
+�
+�
+⟨b2⟩
+and Σ(Γ) = D3(Σ(Λ)) = Σ1212 =
+Example 4.12. Let Λ be the following algebra. Then Σ(Λ) is the following fan by Example 4.5.
+Λ =
+k
+�
+1
+2
+a �
+b
+�
+�
+⟨b2⟩
+Σ(Λ) = Σ2121 =
+Applying Theorem 4.7 to Λ, we get
+Γ :=
+�
+k ⊕ D(ka)
+�
+k
+⟨a,ab⟩k
+�
+0
+⟨e2, b⟩k
+�
+=
+k
+�
+1
+2
+a �
+c
+�
+b
+�
+�
+⟨b2, c2, cab⟩
+and Σ(Γ) = D4(Σ(Λ)) = Σ21312 =
+4.4. Proof of Theorem 1.3. Let k be a field. For a finite dimensional k-algebras Λ of rank 2,
+we regard the g-fan Σ(Λ) as a fan in R2 by isomorphism K0(proj Λ) ≃ R2 given by P1 �→ (1, 0) and
+P2 �→ (0, 1). We denote by
+k-Fan(2)
+the subset of Fansc(2) consisting of g-fans of finite dimensional k-algebras of rank 2. Let k-Fanel(2)
+be the subset of k-Fan(2) consisting of g-fans of finite dimensional elementary k-algebras of rank
+2.
+The following is a main result of this paper.
+Theorem 4.13. For any field k, we have
+k-Fanel(2) = k-Fan(2) = Fansc(2).
+(4.5)
+That is, any sign-coherent fan in R2 can be realized as a g-fan Σ(Λ) of some finite dimensional
+elementary k-algebra Λ.
+Proof. It suffices to show Fansc(2) = k-Fanel(2). Let
+k-Fan+−
+el (2) := k-Fanel(2) ∩ Fan+−
+sc (2) and k-Fan−+
+el (2) := k-Fanel(2) ∩ Fan−+
+sc (2).
+By Gluing Theorem 4.1, we have
+k-Fanel(2) = k-Fan+−
+el (2) ∗ k-Fan−+
+el (2).
+By Rotation Theorem 4.3, k-Fan+−
+el (2) is closed under rotations. By Theorem 4.7 and Proposition
+2.14(b), k-Fan+−
+el (2) is closed under subdivisions. Since Σ(0, 0; 0, 0) = Σ(k × k) ∈ k-Fan+−
+el (2),
+Proposition 2.10 implies
+k-Fan+−
+el (2) = Fan+−
+sc (2).
+Similarly, we have k-Fan−+
+el (2) = Fan−+
+sc (2). Consequently, we have
+Fansc(2)
+(2.5)
+= Fan+−
+sc (2) ∗ Fan−+
+sc (2) = k-Fan+−
+el (2) ∗ k-Fan−+
+el (2) = k-Fanel(2).
+□
+For given Σ ∈ Fansc(2), our proof of Theorem 4.13 gives a concrete algorithm to construct a
+finite dimensional k-algebra Λ satisfying Σ(Λ) = Σ. We demonstrate it in the following example.
+Example 4.14. We construct a finite dimensional k-algebra Γ satisfying Σ(Γ) = Σ13122;1212 by
+the following three steps.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+23
+(I) We obtain a finite dimensional k-algebra
+Λ =
+k
+
+
+1
+2
+a3 �
+a4
+�
+a2
+�
+a1
+�
+
+
+⟨a2
+1, a2
+2, a2
+4, a2a1, a2a3 − a3a4⟩
+satisfying Σ(Λ) = Σ13122;00 by using Rotation Theorem 4.3 and Subdivision Theorem 4.7 as
+follows.
+Σ00
+D2
+�
+Σ111
+D3
+Ex.4.11
+�
+Σ1212
+ρ
+Ex.4.5
+�
+Σ2121
+D4
+Ex.4.12
+�
+Σ21312
+ρ
+Ex.4.6
+�
+Σ13122
+(II) Similarly, we obtain a finite dimensional k-algebra
+Λ′ :=
+k
+�
+1
+2
+b1
+�
+b2
+�
+�
+⟨b2
+2⟩
+satisfying Σ(Λ′) = Σ(0, 0; 1, 2, 1, 2).
+(III) We obtain a finite dimensional k-algebra
+Γ =
+k
+
+
+1
+2
+a3 �
+a4
+�
+a2
+�
+a1
+� b1
+�
+b2
+�
+
+
+⟨a2
+1, a2
+2, a2
+4, a2a1, a2a3 − a3a4, b2
+2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩
+satisfying Σ(Γ) = Σ(1, 3, 1, 2, 2; 1, 2, 1, 2) by applying Gluing Theorem 4.1 to Λ and Λ′, see
+Example 4.2.
+Σ(Λ) =
+Σ(Λ′) =
+Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) =
+4.5. Gluing fans II. In this subsection, we study another type of gluing g-fans. Results in this
+subsection will not be used in the rest of this paper.
+Theorem 4.15. Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive idem-
+potents 1 = e1 + e2 ∈ Λ and 1 = e′
+1 + e′
+2 ∈ Λ′ satisfying e1Λe2 = 0, e′
+1Λ′e′
+2 = 0,
+σ = cone{(0, −1), (1, −1)} ∈ Σ(Λ)
+and σ′ = cone{(1, −1), (1, 0)} ∈ Σ(Λ′).
+(4.6)
+Then, there exists an elementary k-algebra Γ such that
+Σ2(Γ) = (Σ2(Λ) \ {σ}) ∪ (Σ2(Λ′) \ {σ′}).
+Theorem 4.15 is explained by the following picture.
+Σ(Λ) =
+•
++
+−
+?
+P1
+P2
+σ
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+Σ(Λ′) =
+•
++
+−
+!
+P ′
+1
+P ′
+2
+σ′
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+Σ(Γ) =
+•
++
+−
+!
+?
+Q1
+Q2
+❄❄❄❄❄❄
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+
+24
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+The assumption (4.6) is equivalent to that the defining sequences can be written as
+Σ(Λ) = Σ(a1, . . . , an−1, 1; 0, 0) and Σ(Λ′) = Σ(1, b2, . . . , bm; 0, 0).
+In this case, the defining sequence of Σ(Γ) is given by
+Σ(Γ) = Σ(a1, . . . , an−2, an−1 + b2 − 1, b3, . . . , bm; 0, 0).
+The rest of this section is devoted to proving Theorem 4.15. By our assumption, we can write
+Λ =
+�
+A
+X
+0
+B
+�
+and P1 := [A X], P2 := [0 B] ∈ proj Λ,
+Λ′ =
+� C
+Y
+0
+D
+�
+and P ′
+1 := [C Y ], P ′
+2 := [0 D] ∈ proj Λ′,
+where
+• A, B, C, D are local k-algebras such that k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD.
+• X is an Aop ⊗k B-module and Y is an Cop ⊗k D-module.
+• There exist g ∈ X and h ∈ Y such that X = gB ̸= 0 and Y = Ch ̸= 0 by Proposition 3.12.
+The construction of Γ is as follows: Let A ×k C (respectively, B ×k D) be a fibre product of
+canonical surjections A → k and C → k (respectively, B → k and D → k). As in Section 3.2, we
+consider maps
+π : X → X/XJB
+(g(·))−1
+−−−−−→ B/JB = k and π′ : Y → Y/JCY
+((·)h)−1
+−−−−−→ C/JC = k.
+(4.7)
+Let X×kY be a fibre product of π : X → k and π′ : Y → k. Then X×kY is a (A×kC)op⊗k(B×kD)-
+module, and let
+Γ :=
+� A ×k C
+X ×k Y
+0
+B ×k D
+�
+and Q1 := [A ×k C X ×k Y ], Q2 := [0 B ×k D] ∈ proj Γ.
+Consider ideals of Γ by
+I =
+� JC
+JCY
+0
+JD
+�
+and I′ =
+� JA
+XJB
+0
+JB
+�
+.
+Then there exist isomorphisms of k-algebras
+Γ/I ≃ Λ and Γ/I′ ≃ Λ′.
+(4.8)
+As in Section 3.2, for s, t ≥ 0, x ∈ Ms,t(X), y ∈ Ms,t(Y ) and (x′, y′) ∈ Ms,t(X ×k Y ), we define
+Px
+:=
+(P ⊕t
+2
+x(·)
+−−→ P ⊕s
+1 ) ∈ per Λ,
+P ′
+y
+:=
+(P ′
+2
+⊕t
+y(·)
+−−→ P ′
+1
+⊕s) ∈ per Λ′
+Q(x,y)
+:=
+(Q⊕t
+2
+(x′,y′)(·)
+−−−−−−→ Q⊕s
+1 ) ∈ per Γ.
+Proposition 4.16. Let s, t ≥ 0 and (x, y) ∈ Ms,t(X ×k Y ). If Q(x,y) is a presilting complex of Γ,
+then Px is a presilting complex of Λ and P ′
+y is a presilting complex of Λ′.
+Proof. By (4.8) and Q(x,y) ⊗Γ Λ = Px, the complex Px is presilting. The complex P ′
+y is presilting
+similarly.
+□
+Define maps (·) : A → C and (·) : B → D as the compositions of canonical maps
+(·) : A
+(·)
+−→ k ⊂ C and (·) : B
+(·)
+−→ k ⊂ D.
+Using π and π′ in (4.7), define maps (·) : X → Y and (·) : Y → X by
+(·) : X
+π−→ k
+(·)h
+−−→ kh ⊂ Y
+and (·) : Y
+π′
+−→ k
+(·)g
+−−→ kg ⊂ X.
+(4.9)
+Then the first projection X ×k Y → X, (x, y) �→ x has a section given by
+X → X ×k Y, x �→ (x, x),
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+25
+and the second projection X ×k Y → Y , (x, y) �→ y has a section given by
+Y → X ×k Y, y �→ (y, y).
+The following is a crucial result.
+Proposition 4.17. The following assertions hold.
+(a) Let s ≥ t. For x ∈ Ms,t(X), consider (x, x) ∈ Ms,t(X ⊗k Y ). Then Px is a presilting complex
+of Λ if and only if Q(x,x) is a presilting complex of Γ.
+(b) Let s ≤ t. For y ∈ Ms,t(Y ), consider (y, y) ∈ Mc,d(X ⊗k Y ). Then P ′
+y is a presilting complex
+of Λ′ if and only if Q(y,y) is a presilting complex of Γ.
+Proof. It suffices to prove (a) since (b) is dual to (a).
+The “if” part is clear from Proposition 4.16.
+We prove the “only if” part. By Proposition 3.11, it suffices to show
+Ms,t(X ×k Y ) = Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D).
+Since
+X ×k Y = {(0, y) | y ∈ JCY } + {(z, z) | z ∈ X},
+it suffices to show the following assertions.
+(i) For each y ∈ Ms,t(JCY ), we have (0, y) ∈ Ms(A ×k C)(x, x).
+(ii) For each z ∈ Ms,t(X), we have (z, z) ∈ Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D).
+We prove (i). Since Px is presilting, π(x) ∈ Ms,t(k) has full rank by Proposition 3.13. Since s ≥ t,
+the map (·)π(x) : Ms(k) → Ms,t(k) is surjective. Applying JC ⊗k −, the map (·)π(x) : Ms(JC) →
+Ms,t(JC) is also surjective, and so is the composition
+(·)x
+(4.9)
+= (·)π(x)h : Ms(JC)
+(·)π(x)
+−−−−→ Ms,t(JC)
+(·)h
+−−→ Ms,t(JCY ).
+Therefore there exists c ∈ Ms(JC) such that y = cx. Then (0, c) ∈ Ms(A ×k C) satisfies
+(0, c)(x, x) = (0, y).
+We prove (ii). Since Px is presilting, we have Ms,t(X) = Ms(A)x + xMt(B) by Proposition 3.11.
+Thus there exist a ∈ Ms(A) and b ∈ Mt(B) such that z = ax + xb. Then
+(a, a)(x, x) + (x, x)(b, b) = (ax + xb, ax + xb) = (z, z).
+Thus the assertion follows.
+□
+Now we are ready to prove Theorem 4.15.
+Proof of Theorem 4.15. By Proposition 3.5, each of Σ(Λ), Σ(Λ′) and Σ(Γ) contains cone{(−1, 0), (0, 1)}.
+By Propositions 4.16 and 4.17, the following assertions hold.
+(i) Let s ≥ t. Then there exists x ∈ Ms,t(X) such that Px is presilting if and only if there exists
+(x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting.
+(ii) Let s ≤ t. Then there exists y ∈ Ms,t(Y ) such that P ′
+y is presilting if and only if there exists
+(x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting.
+Therefore the claim follows.
+□
+Example 4.18. Let Λ and Λ′ be the following algebras.
+Λ :=
+k
+�
+1
+2
+a �
+b
+�
+�
+⟨b2⟩
+Λ′ :=
+k
+
+
+1
+2
+a �
+d
+�
+c1
+�
+c2
+�
+
+
+⟨c2
+1, c2
+2, d2, c1c2, c1a − ad⟩
+
+26
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+By Examples 4.5 and 4.6, we have
+Σ(Λ) = Σ2121 =
+Σ(Λ′) = Σ13122 =
+Let Γ :=
+� e1Λe1 ×k e1Λ′e1
+e1Λe2 ×k e1Λ′e2
+0
+e2Λe2 ×k e2Λ′e2
+�
+. Then we have
+Γ =
+k
+
+
+1
+2
+a �
+b
+�
+d
+�
+c1
+�
+c2
+�
+
+
+⟨b2, c2
+1, c2
+2, d2, c1c2, c1a − ad, c2ab, bd, db⟩ and Σ(Γ) = Σ214122 =
+5. g-Convex algebras of rank 2
+In this section, we will characterize algebras of rank 2 which have convex g-polygons.
+5.1. Characterizations of g-convex algebras of rank 2. Let e, e′ be pairwise orthogonal prim-
+itive idempotents in A and x ∈ eAe′. Then we use the following notations.
+• x ∈ eAe′ is a left generator (respectively, right generator) of eAe′ if eAx = eAe′ (respectively,
+xAe′ = eAe′).
+• Define subalgebras Lx ⊂ e′Ae′ and Rx ⊂ eAe as follows (see Lemma 5.5).
+Rx := {a ∈ eAe | ax ∈ xAe′} and Lx := {a ∈ e′Ae′ | xa ∈ eAx}.
+Recall that, for an algebra Λ and a right (respectively, left) Λ-module M, we denote by t(MΛ)
+(respectively, t(ΛM)) the minimal number of generators of M.
+Theorem 5.1. Let A be a basic finite dimensional algebra, {e1, e2} a complete set of primitive
+orthogonal idempotents in A and Pi = eiA (i = 1, 2).
+(a) A is g-convex if and only if Σ(A) = Σa;b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.
+(b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)). Then we have the following statements.
+• Σ(A) = Σ00;b for some b if and only if (l, r) = (0, 0).
+• Σ(A) = Σ111;b for some b if and only if (l, r) = (1, 1).
+• Σ(A) = Σ1212;b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some
+left generator x of e1Ae2.
+• Σ(A) = Σ2121;b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some
+right generator x of e1Ae2.
+Σ00;b
+•
+P2
+P1
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+⑧⑧⑧⑧⑧⑧
+Σ111;b
+•
+P2
+P1
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+Σ1212;b
+•
+P2
+P1
+❄
+❄
+❄
+❄
+❄
+❄
+✴✴✴✴✴✴✴✴✴
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+Σ2121;b
+•
+P2
+P1
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+Remark 5.2. For a left (respectively, right) generator x of e1Ae2, Rx (respectively, Lx) is unique
+up to conjugacy. In particular, t(Rxe1Ae1) (respectively, t(e2Ae2Lx)) does not depend on the choice
+of a left (respectively, right) generator x.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+27
+Example 5.3. (a) Here, we give algebras which realize 7 convex g-fans up to isomorphism of
+g-fans. We define Ai = kQ/I (i ∈ {1, 2, 3, 4, 5, 6, 7}) as follows.
+A1 =
+k
+�
+1
+2
+a �
+b�
+c
+�
+d
+�
+�
+⟨ab, ad, ba, bc, c2, d2⟩
+A2 =
+k
+�
+1
+2
+a �
+b�
+d
+�
+�
+⟨ab, ba, d2⟩
+A3 =
+�
+1
+2
+a �
+b�
+d
+�
+�
+⟨ab, ba, ad, d2⟩
+A4 =
+k
+�
+1
+2
+b�
+d
+�
+�
+⟨d2⟩
+A5 =
+k
+�
+1
+2
+a �
+b�
+�
+⟨ab, ba⟩
+A6 = k
+�
+1
+2
+b�
+�
+A7 = k
+�
+1 2
+�
+Then the g-fans Σ(Ai) (i ∈ {1, 2, 3, 4, 5, 6, 7}) are given by the following table.
+i
+1
+2
+3
+4
+5
+6
+7
+Σ(Ai)
+•+
+−
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+✴✴✴✴✴✴✴
+❄❄❄❄❄
+❖❖❖❖❖❖❖
+•+
+−
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄❄❄❄❄
+❄❄❄❄❄
+❖❖❖❖❖❖❖
+•+
+−
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❖❖❖❖❖❖❖
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❖❖❖❖❖❖❖
+•+
+−
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+❄❄❄❄❄
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+❄❄❄❄❄
+•+
+−
+❄❄❄❄❄
+⑧⑧⑧⑧⑧
+⑧⑧⑧⑧⑧
+❄❄❄❄❄
+(b) Let K/k be a field extension with degree two, and A be a k-algebra
+�k
+K
+0
+K
+�
+with e1 =
+�1
+0
+0
+0
+�
+,
+e2 =
+�0
+0
+0
+1
+�
+. We write K = k(t) and set x :=
+�0
+1
+0
+0
+�
+∈ e1Ae2, u =
+�0
+0
+0
+t
+�
+∈ e2Ae2. Then we
+have Lx =
+�0
+0
+0
+k
+�
+, u ̸∈ Lx, and the following equations hold.
+• e1Ae2 =
+�
+0
+K
+0
+0
+�
+= xAe2 = e1Ax + e1Axu
+• e2Ae2 =
+�0
+0
+0
+K
+�
+= Lx + uLx
+Further, we have e2Ae1 = 0. Therefore, Theorem 5.1 implies that Σ(A) has the following form.
+•
+P2
+P1
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄❄❄❄❄❄
+⑧⑧⑧⑧⑧⑧
+5.2. Proof of Theorem 5.1. In this subsection, we prove Theorem 5.1. The following observation
+shows Theorem 5.1(a) and gives another proof of [AHIKM1, Theorem 6.3].
+Proposition 5.4. Let A be as in Theorem 5.1. Then A is g-convex if and only if Σ(A) = Σa;b for
+some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.
+Proof. The “if” part is clear.
+Conversely, assume that A is g-convex and Σ(A) = Σa;b with
+a = (a1, . . . , an) and b = (b1, . . . , bm). Then ai ≤ 2 and bj ≤ 2 hold for each i, j. Using Proposition
+2.10, it is easy to check that a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)} holds (see Figure 1).
+□
+Next we show the following.
+Lemma 5.5. Let x ∈ e1Ae2. Then Lx is a subalgebra of e2Ae2, and Rx is a subalgebra of e1Ae1.
+Proof. This is a special case of the following easy fact: Let A, B be rings, M an (A, B)-module,
+and x ∈ M. Then {b ∈ B | xb ∈ Ax} is a subring of B.
+□
+Now we give a key observation. As in Section 3.2, for s, t ≥ 0, x ∈ Ms,t(e1Ae2), we define
+Px := (e2A⊕t
+x(·)
+−−→ e1A⊕s) ∈ Kb(proj A).
+Proposition 5.6. Assume t(e1Ae1e1Ae2) = 1.
+For a left generator x ∈ e1Ae2, the following
+conditions are equivalent.
+
+28
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+(1) Σ(A) contains cone{(1, −1), (1, −2)}.
+(2) t(Rxe1Ae2) = 2.
+(3) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2) holds for some u ∈ e1Ae1 \ Rx and y := [x ux].
+Proof. Notice that Px is indecomposable presilting by Proposition 3.12.
+(1)⇒(2) If t(Rxe1Ae1) = 1, then e1Ae1 = Rx holds. Thus e1Ae2 = e1Ax ⊂ xAe2 holds, and
+thus x is a right generator. By Proposition 3.12, Px ⊕ P2[1] ∈ 2-siltA holds, a contradiction to
+cone{(1, −1), (1, −2)} ∈ Σ(A). Thus it suffices to prove t(Rxe1Ae1) ≤ 2.
+Since cone{(1, −1), (1, −2)} ∈ Σ(A), there exists y = [x1 x2] ∈ M1,2(e1Ae2) such that Px ⊕ Py
+is silting. By Proposition 3.11, we have
+M1,2(e1Ae2) = e1Ay + yM2,2(e2Ae2),
+(5.1)
+M1,2(e1Ae2) = e1Ay + xM1,2(e2Ae2).
+(5.2)
+Looking at the first entry of (5.1), at least one of x1 and x2 does not belong to rade1Ae1 e1Ae2.
+Without loss of generality, assume x1 /∈ rade1Ae1 e1Ae2. Then there exists a ∈ (e1Ae1)× such that
+x = ax1. Since Py ≃ Pay, we can assume x1 = x by replacing y by ay. Since x is a left generator,
+there exists u ∈ e1Ae1 such that x2 = ux. Consequently, we can assume
+y = [x ux].
+For each a ∈ e1Ae1, (5.2) implies that there exist a′ ∈ e1Ae1 and b, b′ ∈ e2Ae2 such that
+[0 ax] = a′[x ux] + x[b b′].
+Then a′ and a−a′u are in Rx, and hence a = a′u+(a−a′u) ∈ Rxu+Rx. Thus e1Ae1 = Rx +Rxu
+and t(Rxe1Ae1) ≤ 2 hold, as desired.
+(2)⇒(3) Since t(Rxe1Ae2) = 2 and Rx ̸⊂ radRx e1Ae1, there exists u ∈ e1Ae1 \ Rx such that
+Rxu + Rx = e1Ae1.
+Multiplying x from the right, we have Rxux + Rxx = e1Ax = e1Ae2. Since Rxx ⊂ xAe2, we
+have
+Rxux + xAe2 = e1Ae2.
+(5.3)
+To prove (3), take any [z w] ∈ M1,2(e1Ae2). Since x is a left generator, there exists a ∈ e1Ae1
+such that z = ax. By (5.3), there exist r ∈ Rx and b ∈ e2Ae2 such that w − aux = rux + xb. By
+definition of Rx, there exists c ∈ e2Ae2 such that rx = xc. Then we have
+[z w] = (a + r)[x ux] + x[−c b] ∈ e1Ay + xM1,2(e2Ae2).
+(3)⇒(1) By Proposition 3.11, the following assertions hold.
+• Px is presilting if and only if (i) e1Ax + xAe2 = e1Ae2.
+• Py is presilting if and only if (ii) e1Ay + yM2,2(e2Ae2) = M1,2(e1Ae2).
+• HomKb(proj A)(Px, Py[1]) = 0 if and only if (iii) e1Ax + yM2,1(e2Ae2) = e1Ae2.
+• HomKb(proj A)(Py, Px[1]) = 0 if and only if (iv) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2).
+It is clear that (iv) implies (ii), and (i) implies (iii). By looking at the first entry of the row vector,
+(iv) implies (i).
+Our assumption (3) implies that (iv) holds, and hence (i)-(iii) also hold.
+Thus Px ⊕ Py is
+presilting. It remains to show that Py is indecomposable. Suppose that Py is decomposable. By
+considering g-vector, we have that Py ≃ e2A[1] ⊕ Pz for some z ∈ e1Ae2. Since [Pz] = [Px], we
+have Pz ≃ Px by [DIJ, Theorem 6.5(a)]. This shows that e2A[1] ⊕ Px is silting. By Proposition
+3.12, we have xAe2 = e1Ae2 and Rx = eAe. This contradicts u ̸∈ Rx.
+□
+We are ready to prove Theorem 5.1(b).
+Proof of Theorem 5.1(b). The first and second statements follow from Proposition 3.5 and Propo-
+sition 3.12.
+
+FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2
+29
+We prove the third statement. By Proposition 3.12, cone{(1, 0), (1, −1)} ∈ Σ(A) if and only if
+t(e1Ae1e1Ae2) = 1, and cone{(0, −1), (1, −2)} ∈ Σ(A) if and only if t(e1Ae2e2Ae2) = 2. Thus the
+assertion follows from Proposition 5.6.
+The fourth statement is the dual of the third statement.
+□
+Acknowledgments
+T.A is supported by JSPS Grants-in-Aid for Scientific Research JP19J11408. A.H is supported
+by JSPS Grant-in-Aid for Scientists Research (C) 20K03513. O.I is supported by JSPS Grant-
+in-Aid for Scientific Research (B) 16H03923, (C) 18K3209 and (S) 15H05738. R.K is supported
+by JSPS Grant-in-Aid for Young Scientists (B) 17K14169. Y.M is supported by Grant-in-Aid for
+Scientific Research (C) 20K03539.
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+
+30
+TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO
+Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita,
+Osaka 565-0871, Japan
+Email address: [email protected]
+Department of Pure and Applied Mathematics, Graduate School of Information Science and Tech-
+nology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan
+Email address: [email protected]
+Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo
+153-8914, Japan
+Email address: [email protected]
+Department of Information Science and Engineering, Okayama University of Science, 1-1 Ridaicho,
+Kita-ku, Okayama 700-0005, Japan
+Email address: [email protected]
+Faculty of Liberal Arts, Sciences and Global Education / Graduate School of Science, Osaka Met-
+ropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
+Email address: [email protected]
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