https://neckpinch.com/papers/New arXiv papers on Ricci flow and the Poincaré conjecture Thu, 24 Sep 2026 06:48:01 GMT https://arxiv.org/abs/2609.28280arxiv:2609.28280Wed, 23 Sep 2026 00:00:00 GMTLorenzo Foscolo, Mario Garcia-Fernandez — The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.Ricci flowGeneralized & G₂ flowsRicci solitonsEinstein metrics3-manifoldsKähler geometryHomogeneous spacesBoundaries & non-compact https://arxiv.org/abs/2609.27644arxiv:2609.27644Wed, 23 Sep 2026 00:00:00 GMTAndoni Royo Abrego — We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a SU(2) bundle over S^4.Ricci flowSingularitiesPerelman entropy https://arxiv.org/abs/2609.25550arxiv:2609.25550Tue, 22 Sep 2026 00:00:00 GMTPaula Burkhardt-Guim — We prove a Positive Mass Theorem for C^0-asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a C^0-asymptotically flat Riemannian metric that is a C^0 perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a C^0 analog of the classical ADM mass previously introduced by the author.Ricci flowScalar curvatureBoundaries & non-compact https://arxiv.org/abs/2609.24884arxiv:2609.24884Mon, 21 Sep 2026 00:00:00 GMTTristan C. Collins, Genggeng Huang, Freid Tong, Yulun Xu — We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.Ricci solitonsKähler–Ricci solitonsKähler geometry https://arxiv.org/abs/2609.23630arxiv:2609.23630Sun, 20 Sep 2026 00:00:00 GMTYasufumi Nitta — We prove that every extremal Kähler–Ricci soliton on a Fano manifold is Kähler–Einstein. This solves the problem of Calamai and Petrecca in full generality.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometry https://arxiv.org/abs/2609.23424arxiv:2609.23424Sun, 20 Sep 2026 00:00:00 GMTMohammadjavad Habibivostakolaei, Abbas M. Sherif, Yen-Kheng Lim — We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.Ricci solitons4-manifoldsStability & uniquenessPhysics https://arxiv.org/abs/2609.21472arxiv:2609.21472Fri, 18 Sep 2026 00:00:00 GMTLucas Lavoyer, Adam Thompson — We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2609.20513arxiv:2609.20513Thu, 17 Sep 2026 00:00:00 GMTWangjian Jian, Jian Song — The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.Ricci flowKähler–Ricci flowEinstein metricsSingularities(Non)collapsingKähler geometryScalar curvature https://arxiv.org/abs/2609.19724arxiv:2609.19724Thu, 17 Sep 2026 00:00:00 GMTPing-Hung Lee — We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.Ricci flowMean curvature flowSurfacesScalar curvatureDiscrete & graph Ricci https://arxiv.org/abs/2609.18834arxiv:2609.18834Wed, 16 Sep 2026 00:00:00 GMTTongxin Xu, Zhenlei Zhang — We prove that any finite-time collapsing Kähler-Ricci flow on compact Kähler surfaces develops a Type I singularity. Together with the previous results, this implies that any finite time singularity of the Kähler-Ricci flow on compact Kähler surfaces is of Type I.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2609.18594arxiv:2609.18594Wed, 16 Sep 2026 00:00:00 GMTBing Wang, Jie Wang — As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function f admits a heat kernel H_f under the weighted volume measure e^-fdv. In this paper, we study H_f systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between H_f and the spacetime heat kernel H(x,t;y,s) under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that left|∇ Rright|=o(f^3/2) implies R=o(f).Ricci flowRicci solitonsPerelman entropyReduced volumeHeat kernelLimit spaces & RCD https://arxiv.org/abs/2609.18579arxiv:2609.18579Wed, 16 Sep 2026 00:00:00 GMTDaniele Angella, Mauricio Corrêa — Let S be a Kato surface and D its maximal reduced divisor of rational curves. On Ssetminus D we construct Hermitian metrics which are flat along the leaves of the canonical foliation and study their evolution under the Chern-Ricci flow. For Enoki surfaces, we construct an immortal normalised solution which, on every compact sublevel of the natural exhaustion, converges in the Gromov-Hausdorff sense to the elliptic base endowed with an explicit flat metric. For Kato surfaces of intermediate type, the affine Green model and its finite-index extension give, under the assumption 0<μ<2, an explicit normalised solution which, on every compact Green block, collapses in the Gromov-Hausdorff sense to a circle. In the Enoki case the limiting area is 2πb_2(S); in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.Ricci flowGeneralized & G₂ flowsLimit spaces & RCD https://arxiv.org/abs/2609.17183arxiv:2609.17183Tue, 15 Sep 2026 00:00:00 GMTYoussef Ayad — There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group H_2n+1, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra h_2n+1, denoted by μ, ν, and φ. We show that, in a specific case, μ is an algebraic Ricci soliton that is shrinking. Moreover, ν is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra h_3 and it is shrinking. Finally, we show that φ is a steady algebraic Ricci soliton on h_2n + 1 for n > 1. However, for n = 1, φ is flat.Ricci solitons3-manifolds https://arxiv.org/abs/2609.16733arxiv:2609.16733Tue, 15 Sep 2026 00:00:00 GMTCharles Cifarelli, Ronan Conlon, Max Hallgren, Junsheng Zhang — For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2609.15880arxiv:2609.15880Mon, 14 Sep 2026 00:00:00 GMTShubham Dwivedi — We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the 2-sphere thus confirming a conjecture of Branding–Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on S^2. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on ℝP^2, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on S^2 whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz–Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.Ricci flowSurfaces https://arxiv.org/abs/2609.15147arxiv:2609.15147Mon, 14 Sep 2026 00:00:00 GMTBidhan Mondal, Nirabhra Basu, Arindam Bhattacharyya — In this paper, we introduce some type vector fields with respect to a semi-symmetric non-metric (SSNM) connection. We investigate several geometric properties of a K-contact manifold equipped with an SSNM connection and provide a concrete example to justify the relation between the scalar curvature of the SSNM connection and Levi-Civita connection that we have obtained in this paper. Furthermore, we have found the nature of Riemann solitons, conformal Ricci solitons and conformal η-Ricci-Yamabe solitons on K-contact manifolds admitting a SSNM connection.\\ Finally, we determine the necessary and sufficient conditions for such a manifold to be Tildeτ-semi-symmetric, quasi-conformal-semi-symmetric and pseudo-projective-semi-symmetric.Ricci solitonsScalar curvature https://arxiv.org/abs/2609.14866arxiv:2609.14866Mon, 14 Sep 2026 00:00:00 GMTAbdou Bousso, Ameth Ndiaye — We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as R = λ(ξ^flatotimesξ^flat)owedge g. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost θ-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field X is a symmetry of the Ricci tensor of a fixed order k (i.e., L_X^k Ric = 0), the geometric problem reduces to solving a partial differential equation of order k+1 along the flow. Finally, under the assumption that X is a conformal vector field (L_X g = 2φ g) whose infinitesimal flow preserves the line distribution D=Span\ξ\ (with [X,ξ]=aξ for ainℝ), we prove that several key geometric problems (such as establishing the relation L_X^k R = R, determining the minimal order k for X to be a Lie curvature symmetry, or satisfying L_X^k+1R = f L_X^k R for a continuous function f) are equivalent to a scalar differential problem governed by the operator D_X = X + 6φ + 2a.Ricci solitons https://arxiv.org/abs/2609.14580arxiv:2609.14580Sun, 13 Sep 2026 00:00:00 GMTCarlos Daniel Velázquez-Mendoza, María de los Ángeles Sandoval-Romero, Romulo Diaz Carlos — This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.Ricci flowBoundaries & non-compact https://arxiv.org/abs/2609.13938arxiv:2609.13938Sat, 12 Sep 2026 00:00:00 GMTShaochuang Huang, Zhuo Peng — In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.Ricci flow(Non)collapsing3-manifoldsScalar curvatureLimit spaces & RCDStability & uniqueness https://arxiv.org/abs/2609.13052arxiv:2609.13052Fri, 11 Sep 2026 00:00:00 GMTTomoya Tatsuno — We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval [-3, -frac32]. The upper bound -frac32 is achieved by the complex Heisenberg group Heis_3(ℂ) with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold N has the pinching constant -frac32, then N admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant -3. This is derived by showing that there is an open neighborhood U of Heis_3(ℝ)× ℝ^n-3 in the space of n-dimensional 2-step nilmanifolds such that the pinching constant is -3 on U, and any 2-step nilpotent Lie group N has a metric g such that (N,g) lies in U. In fact, if N is not isomorphic to Heis_3(ℝ)× ℝ^n-3, then there is a curve g_t of metrics on N with (N,g_t)in U, showing that there are uncountably many left-invariant metrics on N such that the pinching constant is -3. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant -frac32 is also given, and the pinching constants of various examples are computed.Ricci solitonsCurvature pinchingHomogeneous spacesStability & uniqueness https://arxiv.org/abs/2609.16029arxiv:2609.16029Thu, 10 Sep 2026 00:00:00 GMTHuibin Chen, Zhiqi Chen, Fuhai Zhu — In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions 6m+d for m>d≥1. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra mathfrak m_2(n), n≥5, admits a generalized nilsoliton if and only if 5≤ n≤8.Generalized & G₂ flowsRicci solitonsHomogeneous spaces https://arxiv.org/abs/2609.12237arxiv:2609.12237Thu, 10 Sep 2026 00:00:00 GMTAbdou Bousso, Ameth Ndiaye — In this article, we study Ricci-Yamabe solitons on the Lie group Sol × ℝ^n equipped with a natural left-invariant Riemannian metric, explicitly determining the vector fields that characterize them. We then deduce that, in the case of a Ricci soliton, it is expanding, whereas in the case of a Yamabe soliton, it is shrinking. Finally, we show that if this Lie group is a gradient Ricci-Yamabe soliton, the vector field belongs to Span\∂_t_1, dots, ∂_t_n\, and we explicitly provide the Perelman potential.Ricci solitonsHomogeneous spaces https://arxiv.org/abs/2609.08206arxiv:2609.08206Tue, 08 Sep 2026 00:00:00 GMTJunrong Yan — We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius R, the region where the curvature radius is smaller than R^-1 occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.Ricci flowRicci solitons https://arxiv.org/abs/2609.07124arxiv:2609.07124Mon, 07 Sep 2026 00:00:00 GMTYifan Xiao — In this paper, we study the Kähler–Ricci flow on ℂP^m-bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type I.Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometry https://arxiv.org/abs/2609.06451arxiv:2609.06451Sun, 06 Sep 2026 00:00:00 GMTAlexander Bednarek — We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is n-1, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.Ricci flowKähler–Ricci flowKähler geometryStability & uniqueness https://arxiv.org/abs/2609.04671arxiv:2609.04671Fri, 04 Sep 2026 00:00:00 GMTShuliang Bai, Bobo Hua — For the inverse-weight Lin–Lu–Yau Ricci flow on a finite tree, we prove that the curvature satisfies an exact heat equation and that every edge curvature converges. The limiting curvature vector is independent of the positive initial metric and is the unique minimum-norm point of the base polyhedron of a submodular function determined by the tree; in particular, it can be recovered by finitely many combinatorial minimizations. We also characterize subsequential limits of the normalized weights and obtain criteria for their convergence. Finally, for any two positive initial metrics, the corresponding unnormalized edgewise ratios converge, and the normalized omega-limit sets are related by a projective homeomorphism.Ricci flowDiscrete & graph Ricci https://arxiv.org/abs/2609.03332arxiv:2609.03332Thu, 03 Sep 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Hung Tran — We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds N := N_1 × cdots × N_r, with metric constructed using the ansatz considered in, et. al. In the earlier work by the authors, we considered the "two-bolt" case where both ends of the foliation close with the "bolt" N. In this article, we continue our work on the more subtle "nut-bolt" and "two-nut" cases. The former has one end of the interval closes with a nut-type collapse (i.e. N' := N_2 × cdots × N_r) and the other with a bolt (i.e. N). The compactification widehatM is then a ℂP^m+1-bundle over N'. The "two-nut" case is one that both ends close with nut-type collapses, necessarily two of the N_i's must be ℂP^m_0 and ℂP^m_ℓ, and the compactification widehatM is a ℂP^m_0+m_ℓ+1-bundle over prod_k≥ 3N_k. We proved that in all "two-bolt", "nut-bolt" and "two-nut" caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be (Σ^m+1, g_Σ(t)) × (ℂ^k, textrmflat) with m, k ≥ 0, where Σ is one of the following: ℂP^m+1, textrmTot(L^oplus(m+1)), or a projectivization ℙbig(O^oplus(m_0+1) oplus L^oplus(m_ℓ+1)big) with m_0 + m_ℓ = m, and L is a line bundle over the product of some of the N_1, cdots, N_r factors. The metric g_Σ(t) is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.Ricci flowKähler–Ricci flowRicci solitonsEinstein metricsSingularitiesKähler geometry https://arxiv.org/abs/2609.02878arxiv:2609.02878Wed, 02 Sep 2026 00:00:00 GMTWangjian Jian, Jian Song — We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle X^nrightarrow Y^m, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as \((\C^m,g_{\rm E},J_0)\times(Z',d',J')\). If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to \(\sqrtT-t\). Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder \(\C^m\times\PP^1\).Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2609.01442arxiv:2609.01442Tue, 01 Sep 2026 00:00:00 GMTTongxin Xu, Zhenlei Zhang — We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker ℙ^1× ℂ. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2609.00840arxiv:2609.00840Tue, 01 Sep 2026 00:00:00 GMTJunming Xie — In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.Ricci solitonsKähler–Ricci solitons4-manifoldsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2608.30302arxiv:2608.30302Mon, 31 Aug 2026 00:00:00 GMTXiangsen Qin — Let λ(u,x) be the local Arnold multiplicity of a quasi-plurisubharmonic function u. Di Nezza–Guedj–Lu asked whether every maximal weak solution φ_t of the twisted Kähler–Ricci flow satisfies λ(φ_t,x)=max\λ(φ_0,x)-t,0\. We give counterexamples on the Hirzebruch surface mathbb F_e=mathbb P_mathbb P^1 (mathcal O_mathbb P^1oplusmathcal O_mathbb P^1(-e)), e≥2. Let S be its negative section and F_1,ldots,F_k be distinct fibres. If a,b_i>0, sum_i b_i>ea, and the initial current is a[S]+sum_i b_i[F_i], then λ(φ_t,x)=a-min\k/e,1\t for xin Ssetminusbigcup_iF_i and 0<t<min\a,b_1,ldots,b_k\. Thus the formula fails for k<e; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension n≥2.Ricci flowKähler–Ricci flowKähler geometryWeak & singular flows https://arxiv.org/abs/2608.30027arxiv:2608.30027Sun, 30 Aug 2026 00:00:00 GMTJohn Lott — We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type ℝ^3, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric g(t) converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of (M,t^-1g(t)) is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, s^-1widetilde g(sτ), converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.Ricci flowRicci solitonsGeometrization3-manifoldsLimit spaces & RCDHomogeneous spaces https://arxiv.org/abs/2608.26619arxiv:2608.26619Thu, 27 Aug 2026 00:00:00 GMTKyeongho Bang, Eric Chen, Wenkui Du — We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow (M^2, ∂ M^2, g(t)) on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.Ricci flowAncient solutionsSingularitiesSurfacesStability & uniqueness https://arxiv.org/abs/2608.26598arxiv:2608.26598Thu, 27 Aug 2026 00:00:00 GMTJae Ho Cho — We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE fracddtR=Q(R) in dimensions n=7,8, thereby extending the pinching estimate established by Brendle for n≥ 12 and by Chen for 9≤ n≤ 11. In dimension n=8, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at n=8 by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension n=7, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension n=8. The n=8 pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible (n-1)-dimensional space forms, extending a theorem of Brendle from n≥ 12. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact κ-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from n=4 or n≥ 12.Ricci flowAncient solutions(Non)collapsingCurvature pinchingSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2608.25829arxiv:2608.25829Wed, 26 Aug 2026 00:00:00 GMTJeffrey Streets — Adapting ideas of, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in. Using this we show that low dimensional GRS are diffeomorphic to S^3 / Γ or S^3 × S^1 / Γ. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion G_2, strong torsion Spin(7)-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered 5-manifolds over complex orbifolds.Generalized & G₂ flowsRicci solitons3-manifoldsLimit spaces & RCD https://arxiv.org/abs/2608.25704arxiv:2608.25704Wed, 26 Aug 2026 00:00:00 GMTJeongHyeong Park, Wooseok Shin — We prove that every compact four-dimensional weakly Einstein Ricci soliton is Einstein. The nontrivial compact case reduces to the gradient shrinking setting, where a differential identity for weakly Einstein four-manifolds, together with the curvature identity for gradient Ricci solitons, yields the pointwise relation |R|^2∇ f=0 for the soliton potential f. Consequently, no compact proper weakly Einstein four-manifold admits a Ricci soliton structure. A noncompact homogeneous example shows that the compactness assumption is essential.Ricci solitons4-manifoldsLimit spaces & RCDHomogeneous spacesStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2608.25619arxiv:2608.25619Wed, 26 Aug 2026 00:00:00 GMTElia Fusi, Ramiro A. Lafuente, James Stanfield — We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.Ricci flowGeneralized & G₂ flowsRicci solitonsScalar curvatureHomogeneous spacesStability & uniqueness https://arxiv.org/abs/2608.21732arxiv:2608.21732Sat, 22 Aug 2026 00:00:00 GMTFengyuan Zhang — The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \beginequation D_ijk=\nabla_iR_jk+\nabla_jR_ki+\nabla_kR_ij=0\nonumber \endequation These structures include non-compact gradient shrinking Ricci soliton, compact (m > 1)-quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.Ricci solitonsEinstein metricsBoundaries & non-compact https://arxiv.org/abs/2608.21502arxiv:2608.21502Fri, 21 Aug 2026 00:00:00 GMTBennett Chow, Yuan Liao, Ziyang Qin — We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.Ricci flowSingularities(Non)collapsingCurvature pinching3-manifoldsLimit spaces & RCD https://arxiv.org/abs/2608.20503arxiv:2608.20503Thu, 20 Aug 2026 00:00:00 GMTAnton Khaliapin — We study Ricci solitons on the Riemannian manifold mathbb H^2×mathbb R equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra isom(mathbb H^2×mathbb R). All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.Ricci solitons4-manifolds https://arxiv.org/abs/2608.20040arxiv:2608.20040Thu, 20 Aug 2026 00:00:00 GMTFengjiang Li, Yuanyuan Qu, Guoqiang Wu — Let (M^n, g, f) be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) Ric ≥ frac∇_∇ fRicf on Msetminus D, where D is a compact set over M; (ii) (M^n, g, f) smoothly converges to ℝ^2 × S^n-2, we conclude that (M^n, g, f) is isometric to ℝ^2 × S^n-2. Notably, condition (i) is weaker than the radial flatness condition in.Ricci solitonsScalar curvatureStability & uniqueness https://arxiv.org/abs/2608.19565arxiv:2608.19565Thu, 20 Aug 2026 00:00:00 GMTGanqi Wang, Yongjia Zhang — In this paper, we prove a gap theorem for F-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an ε-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).Ricci flowAncient solutionsScalar curvature https://arxiv.org/abs/2608.19152arxiv:2608.19152Wed, 19 Aug 2026 00:00:00 GMTLongteng Chen, Max Hallgren, Lucas Lavoyer — We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometryBoundaries & non-compact https://arxiv.org/abs/2608.18002arxiv:2608.18002Tue, 18 Aug 2026 00:00:00 GMTTsz-Kiu Aaron Chow, Jingbo Wan — For m=2,3, we prove that every smooth immersion F:ℝℙ^mlooparrowrightmathbb B^,N(1) satisfies κ(F)^2≥ 2m/(m+1), with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with κ(F)≤sqrt3/2 is diffeomorphic to S^3, ℝℙ^3, or S^2× S^1. All three possibilities occur, while κ(F)<sqrt3/2 forces Xcong S^3. These results answer a question of Petrunin and prove a conjecture of Chodosh–Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar–systolic inequality \[ (\min_Y R_g)sys(g)^2<6π^2 \] for every spherical three-space form Y with |π_1(Y)|>2. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar–systolic inequality for ℝℙ^3 of Bray–Brendle–Eichmair–Neves.Ricci flowSurgery3-manifoldsSphere theorems https://arxiv.org/abs/2608.17458arxiv:2608.17458Tue, 18 Aug 2026 00:00:00 GMTTongxin Xu, Zhenlei Zhang — We prove the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow. More precisely, for any compact Kähler manifold Y and its blow-up π:Bl_pYlongrightarrow Y, if [ω_0]-Tc_1(M)=π^*[ω_Y], then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker Tot(O_ℙ^n-1(-1)).Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometry https://arxiv.org/abs/2608.16743arxiv:2608.16743Mon, 17 Aug 2026 00:00:00 GMTAlessandro Goffi, Francesco Pediconi — We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.Ricci flow https://arxiv.org/abs/2608.16248arxiv:2608.16248Mon, 17 Aug 2026 00:00:00 GMTJungwoo Moon — We investigate a compact Einstein-type manifold whose potential vector field generates a Riemannian foliation. In particular, we prove necessary conditions for such a manifold to be taut and to have a splitting property. Additionally, some properties of taut Riemannian foliations on a compact almost Ricci solitons and a compact Einstein manifolds are provided.Ricci solitonsEinstein metrics https://arxiv.org/abs/2608.14478arxiv:2608.14478Fri, 14 Aug 2026 00:00:00 GMTBennett Chow, Robert Koirala — We introduce and study a Fisher information metric \(g^F_τ\) associated to the conjugate heat kernel of a Ricci flow \((M^n,g_t)\). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that \(g^F_τ\) is monotone in scale and satisfies \(g^F_τ\le g_t\). We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect \(g_t-g^F_τ.\) This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities \(0<g^F_τ<g_t\) at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for \(\varphi\)-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to \(0\) to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's \(\varepsilon\)-regularity theorem.Ricci flowPerelman entropyHeat kernelStability & uniqueness https://arxiv.org/abs/2608.14039arxiv:2608.14039Fri, 14 Aug 2026 00:00:00 GMTDongwoo Gang — For a closed connected Riemannian manifold (M,g), the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding xmapsto p_t(x,·)\,dvol_g. The resulting family g_t agrees with Ricci flow to first order in t, but in general not to second order. We prove that g_t =g-2tRic_g +t^2left(-ΔRic_g +2Ric_g^2-frac23Q_gright) +O_C^0(t^3), where Q_g is quadratic in the full curvature tensor. The term -ΔRic_g also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while g_t is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is O(t^2), and round spheres show that this estimate is sharp.Ricci flowHeat kernelLimit spaces & RCD https://arxiv.org/abs/2608.11128arxiv:2608.11128Tue, 11 Aug 2026 00:00:00 GMTRoman Cherniha, John R. King — The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.Ricci flowSurfaces https://arxiv.org/abs/2608.10541arxiv:2608.10541Tue, 11 Aug 2026 00:00:00 GMTTianyin Ren, Chong Song — In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant α=-1 (or +1), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.Ricci flowMean curvature flow https://arxiv.org/abs/2608.08781arxiv:2608.08781Sun, 09 Aug 2026 00:00:00 GMTBeatrice Brienza, Anna Fino, Udhav Fowdar, Gueo Grantcharov — Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a ∇-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact ∇-Einstein manifold in dimension 5 and 7. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with S^1.Ricci flowGeneralized & G₂ flowsEinstein metricsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2608.08533arxiv:2608.08533Sun, 09 Aug 2026 00:00:00 GMTHenry Shin — We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively C^2,α-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder h^2,α neighborhood, a fixed positive-time restart yields the marked first-profile coordinate A_1=λ_∞^-γ_1V_∞in E_1. This amplitude is a split C^1 submersion and locally the projection onto E_1. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.Ricci flowRicci solitonsSingularities4-manifoldsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2608.08370arxiv:2608.08370Sat, 08 Aug 2026 00:00:00 GMTYoussef Ayad — In this paper, we compute the canonical and Kobayashi-Nomizu connections, together with their curvature, on Lorentzian four-dimensional nilpotent Lie groups endowed with a product structure. We also classify the algebraic Ricci solitons associated with these connections.Ricci solitons4-manifoldsHomogeneous spaces https://arxiv.org/abs/2608.08233arxiv:2608.08233Sat, 08 Aug 2026 00:00:00 GMTSergey Stepanov, Irina Tsyganok — Let (M,g) be a compact n-dimensional Riemannian manifold, n>2. We introduce a restricted variational principle for the Einstein-Hilbert functional by requiring the admissible metric variations to satisfy the harmonic gauge condition. We derive the corresponding Euler-Lagrange equation and show that a metric is critical with respect to all volume-preserving harmonic variations if and only if its Einstein tensor differs from a multiple of the metric by an element of the image of the adjoint Bianchi operator. We prove that every harmonic critical metric determines a compact Ricci soliton whose soliton constant is given by the normalized Einstein-Hilbert functional. By Perelman's theorem, every such metric is in fact the metric of a compact gradient Ricci soliton. Conversely, every compact gradient Ricci soliton satisfies the restricted Euler-Lagrange equation. Thus, a compact Riemannian metric is harmonic critical if and only if it is the metric of a compact gradient Ricci soliton. We further show that the gauge one-form differs from the negative differential of a soliton potential by a Killing one-form. In particular, if the Ricci tensor is negative definite, then the gauge one-form vanishes and the metric is Einstein. Moreover, every non-Einstein harmonic critical metric is necessarily shrinking.Ricci solitons https://arxiv.org/abs/2608.07985arxiv:2608.07985Sat, 08 Aug 2026 00:00:00 GMTYeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang — We present a numerical method that constructs geometrically defined coordinates on black-hole horizons, from which the multipole moments are computed without assuming axisymmetry. This method, which we denote the conformal-mapping method (CMM), provides a numerical realization of the conformal construction proposed by Ashtekar et al. in 2022, combining discrete Ricci flow, spectral embedding onto the unit sphere, and Möbius gauge fixing by the vanishing-area-dipole condition. We first test the CMM against analytic Kerr benchmarks, and then apply it to an equal-mass, non-spinning binary black-hole merger. We also compare it with an approximate-symmetry-based method. The CMM allows the multipole moments to be expressed in a fixed reference frame, whereas the symmetry-adapted frame can reorient abruptly when the preferred approximate axis changes. In a frame aligned with the orbital angular momentum, the amplitude of the quadrupole mode grows during inspiral and decays after merger, displaying a qualitative ringdown behavior. These results show that the CMM is a useful tool for studying horizon geometry in dynamical situations where no stable symmetry axis is available.Ricci flowPhysicsDiscrete & graph RicciNumerics & visualization https://arxiv.org/abs/2608.05939arxiv:2608.05939Thu, 06 Aug 2026 00:00:00 GMTShiping Liu, Yunyan Yang — In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the p-Ollivier curvature up to scaling whenever the idleness parameter p≥ 1/2. Moreover, the threshold 1/2 is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).Ricci flowStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2608.02840arxiv:2608.02840Mon, 03 Aug 2026 00:00:00 GMTEric Cochran, Arseny Mingajev, Lawrence Mouillé, Nazia Valiyakath — We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.Ricci flowHomogeneous spacesNumerics & visualization https://arxiv.org/abs/2607.28057arxiv:2607.28057Thu, 30 Jul 2026 00:00:00 GMTGuangwen Zhao — We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton (M,g,J,f) and a real-valued pluriharmonic function u, we investigate conditions under which u must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that u is constant whenever int_M|∇ u|^pdv<∞ for some 0<p<∞. In the shrinking case, we prove the same conclusion for 0<p≤ 2. Finally, we construct a complete Kähler example showing that the extension to the range 0<p<1 relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.Ricci solitonsKähler–Ricci solitonsKähler geometry https://arxiv.org/abs/2607.26240arxiv:2607.26240Tue, 28 Jul 2026 00:00:00 GMTMohammed El Amine Mekki, Ahmed Mohammed Cherif — In this paper, we study Ricci solitons on the three-dimensional solvable Lie group Sol^3_m,n equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical Sol^3 geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from Sol^3_m,n into Euclidean spaces.Ricci solitons3-manifoldsHomogeneous spaces