https://neckpinch.com/topics/curvature-flows/New arXiv papers on other curvature flows, from Neckpinch Wed, 04 Nov 2026 13:15:59 GMT https://neckpinch.com/papers/2611.03249/arxiv:2611.03249Mon, 02 Nov 2026 00:00:00 GMTTianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou — We prove that every smooth closed strictly convex hypersurface in ℝⁿ⁺¹ contracts to a round point under the σₖ-curvature flow for 2≤ k≤ n-1, without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.Other curvature flowsCurvature pinchingLong-time existence & convergence https://neckpinch.com/papers/2611.02674/arxiv:2611.02674Mon, 02 Nov 2026 00:00:00 GMTYa Gao — We study curvature flows for the dual Orlicz–Minkowski problem for strictly convex capillary hypersurfaces in the Euclidean half-space. The prescribed function has the form φ(h)G(X), where X=∇h+hν and G may be nonradial and nonhomogeneous. For acute contact angles, the evenness assumption together with either of the two integral conditions yields a smooth weighted-volume-preserving flow for all t>0 and a smooth solution to the dual Orlicz–Minkowski equation. Without the evenness assumption, two strict spherical-cap barriers yield the same conclusions for an unnormalized flow, whose limiting hypersurfaces solve the dual Orlicz–Minkowski equation with c=1. A uniform strict monotonicity condition under rescaling ensures uniqueness for c=1 and convergence of the unnormalized flow in any C^k norm at an exponential rate. The problem considered here includes the capillary Lₚ-Minkowski and Lₚ dual Minkowski problems as special cases.Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsHomogeneous spacesStability & uniquenessFree boundary & capillary https://neckpinch.com/papers/2611.01968/arxiv:2611.01968Sun, 01 Nov 2026 00:00:00 GMTHaozhao Li, Bing Wang, Kai Zheng — We study global smooth Calabi flows on polarized Kähler surfaces (M,L), assuming uniformly bounded Sobolev constants and the algebraic conditions χ(𝓞_M)=1 and c₁(M)· c₁(L)>0. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and |S|≤1. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ℙ¹×ℙ¹ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.Other curvature flowsKähler geometryScalar curvatureLimit spaces & RCDStability & uniquenessKähler–Einstein & K-stability https://neckpinch.com/papers/2611.01882/arxiv:2611.01882Sun, 01 Nov 2026 00:00:00 GMTHaozhao Li, Bing Wang, Kai Zheng — We study global Calabi flows g(t) on compact Kähler surfaces with a uniform Sobolev constant. For any sequence tⱼ→∞, set gⱼ(t):=g(tⱼ+t). After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime hat C^∞-Cheeger–Gromov topology: it is uniform in the Gromov–Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.Other curvature flowsSingularitiesKähler geometryLimit spaces & RCDKähler–Einstein & K-stability https://neckpinch.com/papers/2611.01623/arxiv:2611.01623Sun, 01 Nov 2026 00:00:00 GMTHongyu Chen, Li Ma, Li Sheng — On every unit-volume flat complex three-torus, we construct a solution of the volume-preserving conformal Chern–Calabi flow with a finite-time Chern Type-I singularity at any prescribed point. The full spatial supremum has an exact asymptotic coefficient, and prescribed rescalings converge to a complete nonflat radial shrinking solution. The positive part of Chern scalar curvature can remain arbitrarily small throughout the flow, as can each prescribed subcritical spatial curvature norm. We determine the punctured terminal metric, its one-point completion and unique nonflat tangent cone, and the terminal energy asymptotics. We also exclude Chern Type-II under a logarithmically improved critical weak L³ bound on Chern scalar curvature, and prove a separate scale-invariant integral criterion involving first Chern–Ricci curvature. The constructed singular solutions satisfy both criteria.Other curvature flowsSingularitiesKähler geometryScalar curvatureKähler–Einstein & K-stability https://neckpinch.com/papers/2610.35707/arxiv: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://neckpinch.com/papers/2610.35394/arxiv: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://neckpinch.com/papers/2610.35064/arxiv:2610.35064Mon, 26 Oct 2026 00:00:00 GMTZe-Yu Ye, Xiao-Liu Wang — In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κ_σ^α, where 0<α≤ 1. Here κ_σ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κ_σ^α. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.Other curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsSelf-shrinkers & translatorsSingularitiesHarnack inequalities https://neckpinch.com/papers/2610.34340/arxiv: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://neckpinch.com/papers/2610.31270/arxiv: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://neckpinch.com/papers/2610.31068/arxiv: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://neckpinch.com/papers/2610.30771/arxiv:2610.30771Thu, 22 Oct 2026 00:00:00 GMTJinyu Guo, Liangjun Weng — In this paper, we study the quermassintegral inequalities for capillary hypersurfaces in a horoball in hyperbolic space. We construct a locally constrained nonlinear curvature flow that preserves the k-th quermassintegral while monotonically increasing the l-th quermassintegral for any (1≤ l<k≤ n). Using this flow, we establish a full family of the quermassintegral inequalities for horo-convex capillary hypersurfaces in a horoball with contact angle θ∈(0,π/2]. Moreover, we weaken the horo-convexity to convexity and obtain the quermassintegral inequalities for convex capillary hypersurfaces between the n-th and l-th quermassintegrals.Other curvature flowsVolume-preserving & constrained flowsFree boundary & capillaryGeometric inequalities https://neckpinch.com/papers/2610.29852/arxiv:2610.29852Wed, 21 Oct 2026 00:00:00 GMTWeimin Sheng, Ye Zhu — We give a finite negative-shift criterion for preserving a prescribed exterior noncollapsing bound under fully nonlinear curvature flows whose hypersurfaces need not be convex. The criterion uses the optimized two-point expression of Andrews and Langford. We verify it by a uniform global two-matrix estimate for the explicit speed F_ε(A)= tr A- εfracnormA^(∘)² tr A on the open cone Γ=A∈Sym(n): tr A>0, normA^(∘)< tr A\. Here A^∘ denotes the trace-free part of A. This cone contains the positive cone and also matrices with a negative eigenvalue, while the displayed speed is concave and non-convex. For an explicit range of parameters depending on the initial exterior bound, the cone and that bound are preserved throughout the smooth embedded existence interval. Together with the known interior estimate for concave speeds, this gives two-sided tangent-ball bounds for this nonlinear family, including closed nonconvex initial hypersurfaces.Other curvature flows(Non)collapsingEigenvalues & spectrum https://neckpinch.com/papers/2610.25745/arxiv:2610.25745Mon, 19 Oct 2026 00:00:00 GMTYuerong Bian, Xu Xu — We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the L^∞ norm.Other curvature flowsKähler geometryStability & uniquenessHarmonic maps & DeTurck trickDiscrete & graph Ricci https://neckpinch.com/papers/2610.18761/arxiv:2610.18761Tue, 13 Oct 2026 00:00:00 GMTXinxiu Zhang — We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ℍⁿ⁺¹, n≥ 2, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the C^(1,α)∩ W^(2,k+1) approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For 0≤ k≤ n-1, we construct weak flows preserving the k-th modified quermassintegral Wₖ with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of Wₖ₊₁ for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.Other curvature flowsWeak & singular flowsStability & uniquenessGeometric inequalities https://neckpinch.com/papers/2610.18423/arxiv: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://neckpinch.com/papers/2610.08270/arxiv:2610.08270Tue, 06 Oct 2026 00:00:00 GMTTianhong Pu, Xiaoqian Xin, Lixia Yuan, Xiaoyu Zhang — We study the graphical forced mean curvature flow in a planar strip subject to Robin boundary conditions. The normal velocity is prescribed by V=H+b, where b>0 is constant and H denotes the curvature of the graph. We consider positive even solutions and reduce the problem to the half interval. The main results describe the degeneration produced by the Robin condition: the height, the instantaneous velocity, and the spatial gradient diverge, while the normalized slope uₓ/u converges to one away from the symmetry axis. The normalized velocity uₜ/u converges to the forcing parameter b. The curvature converges to zero in the interior in an integral sense, whereas its horizontal curvature measure concentrates at the symmetry axis.Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsBoundaries & non-compact https://neckpinch.com/papers/2610.05909/arxiv: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://neckpinch.com/papers/2610.05217/arxiv: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://neckpinch.com/papers/2609.31023/arxiv: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://neckpinch.com/papers/2609.29199/arxiv:2609.29199Thu, 24 Sep 2026 00:00:00 GMTHongyi Sheng, Weimin Sheng, Jiazhuo Yang — Let H^n+1 be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^betar; we treat both regimes: beta>1+kalpha and beta=1+kalpha. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_infty respectively in two different cases.Other curvature flowsHomogeneous spaces https://neckpinch.com/papers/2609.27444/arxiv:2609.27444Wed, 23 Sep 2026 00:00:00 GMTZhixin Wang — We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.Mean curvature flowOther curvature flowsInverse mean curvature flowSingularities3-manifoldsPhysics https://neckpinch.com/papers/2609.24431/arxiv: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://neckpinch.com/papers/2609.13829/arxiv: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://neckpinch.com/papers/2609.01346/arxiv:2609.01346Tue, 01 Sep 2026 00:00:00 GMTHaoxuan Cheng, Junqi Lai, Guoxin Wei — For every pair of integers 3≤ k<n with k odd, we construct a compact embedded rotational torus in ℝⁿ⁺¹ whose homothetic dilations satisfy the unnormalised σₖ-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity C^(1,1/k) and Sobolev regularity W^(2,p) for every 1≤ p<k/(k-1). Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation ⟨ X,ν⟩=−σₖ, where X is the position vector and ν is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical C² rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.Other curvature flowsRotational symmetry & warped products https://neckpinch.com/papers/2608.27093/arxiv:2608.27093Thu, 27 Aug 2026 00:00:00 GMTFábio Nunes da Silva, Edwin Salinas Reyes — We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. We prove that for each vector v, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.Curve shorteningOther curvature flowsInverse mean curvature flow3-manifoldsSurfacesPhysics https://neckpinch.com/papers/2608.22494/arxiv:2608.22494Sun, 23 Aug 2026 00:00:00 GMTXinqun Mei, Guofang Wang, Liangjun Weng — We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.Other curvature flowsGauss curvature flowStability & uniquenessFree boundary & capillary https://neckpinch.com/papers/2608.20037/arxiv: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://neckpinch.com/papers/2608.17315/arxiv:2608.17315Tue, 18 Aug 2026 00:00:00 GMTXiaoshang Jin, Yao Wan, Jie Xiao — This paper establishes sharp upper bounds for p-capacities Cap_(1<p<∞) in the hyperbolic space ℍⁿ through hyperbolic quermassintegrals and effective curvature radii. The quermassintegral comparisons involve Wₙ₋₁, Wₖ₊₁+k(n+1-k)⁻¹Wₖ₋₁, and the pair W₁mid W₂. For star-shaped, mean-convex or h-convex hypersurfaces, inverse mean curvature flow further produces curvature radii determined by L^q-averages of the normalized mean curvature and by moments of its squared hyperbolic excess. These radii convert the resulting estimates into sharp geodesic-ball comparisons for the capacity-to-area ratio. In the range p>2m+1, an interpolating radius combines the 2m-th curvature-excess radius with the L^∞ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalities https://neckpinch.com/papers/2608.11841/arxiv:2608.11841Wed, 12 Aug 2026 00:00:00 GMTChaoqun Gao, Yong Wei, Rong Zhou — We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.Mean curvature flowOther curvature flowsInverse mean curvature flowEigenvalues & spectrum