https://neckpinch.com/topics/long-time-existence/New arXiv papers on long-time existence & convergence, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.35707arxiv:2610.35707Mon, 26 Oct 2026 00:00:00 GMTTianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou — We study quermassintegral-preserving curvature flows in ℍⁿ⁺¹, n≥3, driven by the curvature quotient (Eₖ/E_ℓ)^(1/(k-ℓ)). For 1≤ℓ<k with k+ℓ≤ n, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurface with positive sectional curvature, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. As an application, we establish sharp Alexandrov–Fenchel inequalities between Wₖ and W_ℓ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever 0≤ℓ<k≤ n and k+ℓ≤ n. Equality holds if and only if the domain is a geodesic ball. Together with previously known cases, these inequalities give the full family of Alexandrov–Fenchel inequalities for this class of domains in ℍ⁶. We also construct counterexamples showing that k+ℓ≤ n is the optimal range for preservation of positive sectional curvature by these flows.Other curvature flowsSphere theoremsLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2610.35394arxiv:2610.35394Mon, 26 Oct 2026 00:00:00 GMTJunfang Li — We extend the quermassintegral inequalities of our earlier work from star-shaped domains to a substantially broader class of domains in Euclidean space, which we call spiral-star-shaped. A smooth domain is spiral-star-shaped if its boundary satisfies ⟨ X+A X,ν⟩>0 for some fixed skew-symmetric linear map A, where X is the position vector and ν is the outward unit normal. The vector field X+A X is conformal Killing, with dilation component X and rotational component A X; its trajectories combine expansion and rotation, giving rise to the spiral terminology. When A=0, it reduces to ordinary star-shapedness, while nonzero A allows domains that are not star-shaped in the usual radial sense. For smooth k-convex spiral-star-shaped domains, we prove long-time existence and exponential convergence to a round sphere for a locally constrained inverse curvature flow. The positive spiral support function provides the geometric control needed to treat initial hypersurfaces that need not be star-shaped in the usual sense. Monotonicity of the quermassintegrals along the flow yields sharp quermassintegral inequalities in this enlarged class, with equality if and only if the domain is a ball.Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2610.34665arxiv:2610.34665Mon, 26 Oct 2026 00:00:00 GMTYuan Liao — We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy U(2), except at one finite time when both groups are U(1)²; its smooth limit has holonomy SU(2). The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.Ricci flowGeneralized & G₂ flowsSingularitiesLong-time existence & convergence https://arxiv.org/abs/2610.34340arxiv:2610.34340Sun, 25 Oct 2026 00:00:00 GMTJunhao Tian, Kai Zheng — In this paper, we prove the long time existence of the pseudo-Calabi flow on a compact toric variety for any Tⁿ-invariant initial Kähler metric. If the Mabuchi functional is proper modulo automorphisms, the flow converges exponentially to a constant scalar curvature Kähler (cscK) metric. In particular, this gives a flow proof that properness of the K-energy implies the existence of a cscK metric in the toric setting.Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.31270arxiv:2610.31270Thu, 22 Oct 2026 00:00:00 GMTHongyu Chen, Li Ma, Li Sheng — We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension n≥3. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to L² in time. A uniform curvature bound in the evolving L^p norm, for one fixed p>n, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial L^p curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern–Ricci form gives exponential convergence on Chern–Ricci-flat Hermitian and positive Kähler–Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.Other curvature flowsEinstein metricsRicci-flat & Calabi–YauSingularitiesKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.31068arxiv:2610.31068Thu, 22 Oct 2026 00:00:00 GMTLi Ma, Li Sheng — In this paper, we propose the Chern-Calabi Flow for extremal metrics in conformal Hermitian geometry. After the existence of extremal metric, we consider the existence of global flows to the Chern-Calabi flows for extremal metrics on compact complex surfaces with Gauduchon metrics. To achieve this goal, we use the Adams inequality and the Gronwall inequality method to obtain fourth order derivative norm estimates. We also study the convergence of the global solution to the Chern-Calabi flow and its limit is the solution of the Chern-Calabi extremal metric problem. We propose a fourth order elliptic problem and derive the Bochner formula relevant to the elliptic estimates for the Chern-Calabi heat flow.Other curvature flowsKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.30939arxiv:2610.30939Thu, 22 Oct 2026 00:00:00 GMTYu Li — We study ancient Ricci flows whose tangent flow at backward infinity is a smooth compact Ricci shrinker. After normalization, every nontrivial solution either converges exponentially or has polynomial decay governed by neutral directions. We recover the finite-dimensional parametrization of the exponentially convergent solutions and establish finite-order asymptotic expansions for all slowly convergent solutions. For principal slow solutions, we obtain a locally bijective finite-dimensional parametrization compatible with diffeomorphisms, parabolic rescaling, and time translation.Ricci flowRicci solitonsAncient solutionsSingularitiesLong-time existence & convergence https://arxiv.org/abs/2610.21884arxiv:2610.21884Thu, 15 Oct 2026 00:00:00 GMTKurando Baba, Naoyuki Koike — Principal orbits of cohomogeneity one Hermann actions form a natural class of equifocal hypersurfaces in symmetric spaces of compact type. Based on Kollross's classification of cohomogeneity one actions on irreducible symmetric spaces of compact type, we obtain the time evolution of the squared norm of the shape operator and the mean curvature along the long-time solution of the backward mean curvature flows starting from their principal orbits. These results explicitly describe the authors' previous generalization of the results of Liu-Terng on backward mean curvature flows of isoparametric hypersurfaces to this class of equifocal hypersurfaces.Mean curvature flowHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2610.20265arxiv:2610.20265Wed, 14 Oct 2026 00:00:00 GMTEsther Cabezas-Rivas, Carlo Sinestrari — We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.Mean curvature flowVolume-preserving & constrained flowsSphere theoremsStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2610.19273arxiv:2610.19273Tue, 13 Oct 2026 00:00:00 GMTYifan Guo, Xiaokang Wang — We study the α-Ricci flow (g(t),α(t)), which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus M. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the α-Ricci flow depends on the cohomology class [α] ∈ H¹(M,ℝ), which is invariant under the flow. If [α] is rational, then as t→∞, (M,t⁻¹g(t)) Gromov–Hausdorff converges to S¹ with length ℓ([α])/sqrt 2 where ℓ([α]) is the primitive length of [α]. On the universal cover, the rescaled flow (s⁻¹tilde g(st),tildeα(st)) smoothly converges to an expander (g^*(t),α^*(t)) on ℝ² as s→ ∞ where α^*(t) is nonzero and constant in spacetime and g^*(t) is flat and expanding in the α^* direction. If [α] is irrational, then there is a sequence sᵢ→∞ such that (M,sᵢ⁻¹g(sᵢt)) Gromov–Hausdorff converges to a point. Moreover, on the universal cover, (sᵢ⁻¹tilde g(sᵢt),tildeα(sᵢt)) smoothly converges to (g^*(t),α^*(t)).Ricci flowRicci solitonsLimit spaces & RCDLong-time existence & convergence https://arxiv.org/abs/2610.18423arxiv:2610.18423Tue, 13 Oct 2026 00:00:00 GMTXinxiu Zhang — In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted α-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any α>0 the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsLong-time existence & convergence https://arxiv.org/abs/2610.14227arxiv:2610.14227Fri, 09 Oct 2026 00:00:00 GMTHilário Alencar, Márcio Batista, Gregório Silva Neto — For 1≤ r≤ n-1 we study shrinking self-similar ends of the r-mean curvature flow, Sᵣ=c⟨ X,N⟩ with c<0, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect xu'−u is strictly negative, which yields r-admissibility, and the profile has a complete asymptotic expansion for u(x)/x in powers of x^(−(r+1)), valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly r-admissible cones whose links have constant principal curvatures.Mean curvature flowStability & uniquenessBoundaries & non-compactLong-time existence & convergenceEigenvalues & spectrum https://arxiv.org/abs/2610.05909arxiv:2610.05909Mon, 05 Oct 2026 00:00:00 GMTFuya Hiroi — We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles θ_±, the angle θ between the skew lines, and the initial data γ₀, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.Other curvature flowsVolume-preserving & constrained flowsStability & uniquenessBoundaries & non-compactFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2610.05217arxiv:2610.05217Sun, 04 Oct 2026 00:00:00 GMTHaozhao Li, Bing Wang, Kai Zheng — We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.Other curvature flowsSingularitiesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.03795arxiv:2610.03795Thu, 01 Oct 2026 00:00:00 GMTZhiqi Chen, Jianhua Yu, Huibin Chen, Junna Ni — Let G be a compact connected semisimple Lie group, let Q be a bi-invariant metric and let ρ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with [H₀]=[ρ] exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form [H₀]=[ρ] and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauScalar curvatureHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2610.00732arxiv:2610.00732Wed, 30 Sep 2026 00:00:00 GMTAndrea Poiatti — The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.Mean curvature flowWeak & singular flowsLong-time existence & convergencePhase fields & multiphase flows https://arxiv.org/abs/2609.34923arxiv:2609.34923Mon, 28 Sep 2026 00:00:00 GMTCarlos Alberto Ochoa Flores — A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauLong-time existence & convergence https://arxiv.org/abs/2609.34108arxiv:2609.34108Mon, 28 Sep 2026 00:00:00 GMTSoichiro Uemura — We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.Ricci flowLong-time existence & convergenceDiscrete & graph Ricci https://arxiv.org/abs/2609.31023arxiv:2609.31023Fri, 25 Sep 2026 00:00:00 GMTHongyi Sheng, Weimin Sheng, Jiazhuo Yang — Let S^n+1_+ be the open hemisphere of the unit sphere S^n+1 centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k alpha and the critical regime beta=1+kalpha, where beta is the growth order of the profile at the origin, f(r) almost equals r^beta as r descends to 0. Under the structural condition that f^{1/(1+kalpha)} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_infty>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.Other curvature flowsHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2609.24431arxiv:2609.24431Mon, 21 Sep 2026 00:00:00 GMTZhishuai Liu, Guoxin Wei — In this paper, we study the anisotropic-length-preserving weighted anisotropic curvature flow for smooth convex closed plane curves. For any smooth, embedded, closed, convex initial curve, the flow exists smoothly for all time, and the anisotropic curvature converges smoothly to that of the boundary of the corresponding homothetic Wulff shape as time tends to infinity.Other curvature flowsAnisotropic & nonlocal flowsLong-time existence & convergence https://arxiv.org/abs/2609.23627arxiv:2609.23627Sun, 20 Sep 2026 00:00:00 GMTNaotoshi Fujihara, Toru Kajigaya, Albert Wood — We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies ρ= Cω+ ndd^cf, and which satisfy C>0. With analogous assumptions on the flow, we obtain subconvergence to an f-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that (M,g,f) is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.Mean curvature flowCurve shorteningLagrangian mean curvature flowEinstein metricsSingularitiesSurfacesKähler geometryHomogeneous spacesLong-time existence & convergenceKähler–Einstein & K-stability 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 flowLagrangian mean curvature flowSurfacesScalar curvatureLong-time existence & convergenceDiscrete & graph Ricci 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 S∖ 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 2πb₂(S); in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.Ricci flowGeneralized & G₂ flowsLimit spaces & RCDLong-time existence & convergence https://arxiv.org/abs/2609.18378arxiv:2609.18378Wed, 16 Sep 2026 00:00:00 GMTYuyue Chen, Shiying Wu — We establish the eventual monotonicity of the Willmore deficit ∫_ΣH² dμ−16π along the mean curvature flow of smooth closed embedded strictly convex surfaces in ℝ³, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.Mean curvature flow3-manifoldsSphere theoremsLong-time existence & convergenceGeometric inequalitiesPhysics 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 S². 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 S² 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 flowSurfacesLong-time existence & convergence https://arxiv.org/abs/2609.13829arxiv:2609.13829Sat, 12 Sep 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsCurvature pinchingStability & uniquenessLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2609.10197arxiv:2609.10197Wed, 09 Sep 2026 00:00:00 GMTAprameya Girish Hebbar, Guanhua Shao, Qi Sun — We show that on any closed orientable real-analytic Riemannian surface, any smoothly immersed closed immortal curve-shortening flow converges to a unique non-constant closed geodesic.Curve shorteningLong-time existence & convergence 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 ℝ³, 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 spacesLong-time existence & convergence https://arxiv.org/abs/2608.20037arxiv:2608.20037Thu, 20 Aug 2026 00:00:00 GMTDake Li, Zhizhang Wang, Shiqi Yin — This paper studies self-shrinkers and the long-time behavior of the inverse σₖ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse σₖ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its σₖ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.Other curvature flowsSelf-shrinkers & translatorsBoundaries & non-compactLong-time existence & convergencePhysics https://arxiv.org/abs/2608.10745arxiv:2608.10745Tue, 11 Aug 2026 00:00:00 GMTSara Albert-Niclòs, Esther Cabezas-Rivas, Shujing Pan — We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential C^∞ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.Other curvature flowsCurvature pinchingHomogeneous spacesLong-time existence & convergenceGeometric inequalities