The quermassintegral inequalities for k-convex spiral-star-shaped domains

Mon, 26 Oct 2026 00:00:00 GMT

Junfang 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.

Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

Mon, 26 Oct 2026 00:00:00 GMT

Ze-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.

Quermassintegral inequalities for capillary hypersurfaces in a horoball

Thu, 22 Oct 2026 00:00:00 GMT

Jinyu 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

Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow

Wed, 14 Oct 2026 00:00:00 GMT

Esther 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.

Curvature measures and volume preserving flows in h-convex geometry

Tue, 13 Oct 2026 00:00:00 GMT

Xinxiu 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.

Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

Tue, 06 Oct 2026 00:00:00 GMT

Tianhong 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.

Area-preserving curvature flow for open curves with general contact angles on skew lines

Mon, 05 Oct 2026 00:00:00 GMT

Fuya 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.

Alexandrov–Fenchel inequalities for convex domains in the sphere

Sat, 12 Sep 2026 00:00:00 GMT

Tianci 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.

De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

Sat, 04 Jul 2026 00:00:00 GMT

Tim Laux, Andrea Poiatti — We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate L² distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

Area-preserving crystalline curvature flow in two dimensions

Fri, 05 Jun 2026 00:00:00 GMT

Eric Kim — We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on φ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, i.e. without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz φ-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in. The novelty of our approach lies in the application of φ-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.

Constrained Curvature Flows on Pinched Hadamard Surfaces

Wed, 15 Apr 2026 00:00:00 GMT

Sara Albert-Niclòs, Esther Cabezas-Rivas — We study area- and length-preserving curvature flows for embedded closed curves on pinched Hadamard surfaces. In the variable-curvature setting, the evolution equations contain additional lower-order terms, so the PDE analysis requires refined comparison arguments and delicate curvature estimates. For smooth convex initial curves, we prove preservation and instantaneous strictness of convexity, long-time existence, and uniform bounds for the curvature and its higher derivatives. Under additional geometric assumptions, we obtain convergence of the curvature to a constant. In the rotationally symmetric case, the area-preserving flow exhibits a dichotomy: either the evolving curves converge exponentially to a geodesic circle, or they drift off to infinity and approach a constant-curvature limit curve. We also identify a geometric condition on the initial curve that prevents escape to infinity and guarantees convergence to a geodesic circle.

On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

Mon, 13 Apr 2026 00:00:00 GMT

Zhishuai Liu, Guoxin Wei — In this paper, we study the area-preserving and length-preserving κ^α-type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane ℍ² for α<0 and prove that convexity is preserved along the flows. Assuming that the flows exist for all time, we show that the evolving curves converge smoothly to geodesic circles. Furthermore, we also derive a sufficient condition for global existence of the flows.

Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

Wed, 25 Mar 2026 00:00:00 GMT

Vedansh Arya, Seongmin Jeon, Vesa Julin — The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.

Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

Sun, 08 Feb 2026 00:00:00 GMT

Linlin Fan, Peibiao Zhao — Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle θ∈(0,π) and confirmed the existence of solutions by the standard PDE theory. In the present paper, we study a fractional mean curvature flow for C^(1,1)-regular hypersurfaces with a capillary-type boundary condition and obtain the short time existence by the fixed point argument.

Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

Mon, 22 Dec 2025 00:00:00 GMT

Kuicheng Ma — In this paper, we study the behavior of some locally constrained inverse curvature flow in de Sitter space, with initial value any closed spacelike k-convex hypersurface satisfying some pinching condition. Assume further the Heintze-Karcher inequality for any closed spacelike mean convex hypersurface in de Sitter space, we derive a class of Alexandrov-Fenchel inequalities.

Parabolic free boundary phase transition and mean curvature flow

Tue, 16 Dec 2025 00:00:00 GMT

Jingeon An-Lacroix, Kiichi Tashiro — It is known that there is a strong relation between the parabolic Allen–Cahn equation and the mean curvature flow, in the sense that the parabolic Allen–Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow v=−H-∂_νlog |∇u|+f(u)/|∇u|, satisfied by level surfaces of any solution to the nonlinear parabolic equation ∂ₜu=Δu-f(u). Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen–Cahn equation \{beginalignedat2 ∂ₜu&=Δu &&in \|u|<1\ |∇u|&=1/ε &&on ∂\|u|<1\, endalignedat and confirm that under reasonable assumptions, the C^α norm of the forcing term ∂_νlog|∇u| converges to zero at an algebraic rate as ε→ 0, uniformly in time. This implies that the parabolic free boundary Allen–Cahn equation converges to the mean curvature flow, uniformly (in ε and in time) in the C^(2,α) sense.

A Dynamical Approach to the Berezin-Li-Yau Inequality

Mon, 01 Dec 2025 00:00:00 GMT

Anton Alexa — We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean R_Λ(Ωₜ). For convex domains we show that R_Λ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density Q_Λ and the mean curvature H, established in all dimensions: in d=2 via a near-disk Fourier analysis, and in d≥ 3 via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Cesàro-Pólya inequality for eigenvalue averages.

Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space

Sat, 01 Nov 2025 00:00:00 GMT

Yaoting Gui, Yuqiao Li, Jun Sun — In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.

Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

Sun, 28 Sep 2025 00:00:00 GMT

Jingshi Cui, Peibiao Zhao — In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold M for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature W of M satisfies W ∈ (-∞, W₀ )∪ \ 0\ ∪ (W₀, +∞), where W₀ <0 and W₀ >0 are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when W ≥ 0, wherea it is a geodesic of nonvanishing curvature when W is a negative constant. Specially, in the first Heisenberg group M(0), we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in M(0) and provide a new proof of the fact that the total curvature of γ⊂ M(0) with strictly positive curvature equals 2π.

Existence of flat flows for volume-preserving mean curvature flow with contact angle

Tue, 23 Sep 2025 00:00:00 GMT

Jiwoong Jang — We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we establish the local-in-time equi-boundedness of approximate solutions and a uniform L²-estimate of multipliers. The difficulty is handled by conducting blowup analysis at a point in contact to a spherical cap with sharp angle.

Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

Tue, 12 Aug 2025 00:00:00 GMT

Fabius Krämer — The famous thresholding scheme by Merriman, Bence, and Osher (Motion of multiple junctions: A level set approach. Journal of Computational Physics 112.2 (1994): 334-363.) proved itself as a very efficient time discretization of mean curvature flow. The present paper studies a multiphase, volume constrained version on a weighted manifold that naturally arises in the context of data science. The main result of this work proves convergence to multiphase, weighted, volume-preserving mean curvature flow on a smooth, closed manifold. The proof is only conditional in the sense that convergence of the approximating energy to the weighted perimeter is assumed. These type of assumptions are natural and common in the literature. The proof shows convergence in the energy dissipation inequality, induced by the underlying gradient flow structure, similar to the work of Laux and Otto (The thresholding scheme for mean curvature flow and De Giorgi's ideas for minimizing movements. The Role of Metrics in the Theory of Partial Differential Equations 85 (2020): 63-94.). This leads to a new De Giorgi solution concept that describes weakly the limiting flow. Estimates regarding the derivatives of the heat kernel are developed in order to pass to the limit in the variations of the energy and metric.

A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

Tue, 05 Aug 2025 00:00:00 GMT

Andrea Kubin, Domenico Angelo La Manna, Enrico Pasqualetto — In this article, we introduce a variational algorithm, in the spirit of the minimizing movements scheme, to model the volume-preserving anisotropic mean curvature flow in 2D. We show that this algorithm can be used to prove the existence of classical solutions. Moreover, we prove that this algorithm converges to the global solution of the equation.

Shrinkers of the area-preserving curve-shortening flow: Existence and saddle-point property

Fri, 01 Aug 2025 00:00:00 GMT

Nikita Cernomazov — We consider homothetic evolutions of the area-preserving curve-shortening flow (APCSF), that is, classical curve shortening flow with an additional non-local forcing term. By using known results on λ-curves, we prove the existence of non-circular shrinkers for this flow. In our first main result, we present a partial classification scheme, similar to the well-known Abresch-Langer classification for shrinkers of curve-shortening flow. Finally, we also deduce a saddle-point property for all non-circular (APCSF)-shrinkers analogous to the known saddle-point property of Abresch-Langer curves.

On an area-preserving inverse curvature flow for plane curves

Wed, 30 Jul 2025 00:00:00 GMT

Zezhen Sun, Yuting Wu — In this paper, we study a 1/κⁿ-type area-preserving non-local flow of convex closed plane curves for any n>0. We show that the flow exists globally, the length of evolving curve is non-increasing, and the limiting curve will be a circle in the C^(∞) metric as time t→∞.

A locally constrained inverse Hessian quotient flow in de Sitter space

Tue, 15 Jul 2025 00:00:00 GMT

Kuicheng Ma — In this paper, an Alexandrov-Fenchel inequality is established for closed 2-convex spacelike hypersurface in de Sitter space by investigating the behavior of the locally constrained inverse curvature flow \beginalign \frac{\partial }{\partial t}x=\bigg(u-\fracλ'E_1E_2\bigg)ν,\nonumber \endalign which provides a partial answer to the conjecture raised by Hu and Li in.

Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

Fri, 11 Jul 2025 00:00:00 GMT

Andrea Poiatti — In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel-Laux [J. Differential Geom. 130, 209-268 (2025)] for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao [Arch. Ration. Mech. Anal. 247, 52 (2023)], any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Sat, 28 Jun 2025 00:00:00 GMT

Jie Wu — In this paper, we establish a broad class of new sharp Alexandrov-Fenchel inequalities involving general convex weight functions for static convex hypersurfaces in hyperbolic space. Additionally, we derive new weighted Minkowski-type inequalities for static convex hypersurfaces in hyperbolic space ℍⁿ and for convex hypersurfaces in the sphere Sⁿ. The tools we shall use are the locally constrained inverse curvature flows in hyperbolic space and in the sphere.

Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

Thu, 29 May 2025 00:00:00 GMT

Andrea Chiesa, Keisuke Takasao — In this paper, we propose a new notion of Brakke inequality for volume preserving mean curvature flow. We show the existence of integral varifolds solving the flow globally-in-time in the corresponding Brakke sense using the phase field method. Moreover, such varifolds are solutions to volume preserving mean curvature flow in the L²-flow sense as well. We thus extend a previous result by one of the authors [25].

A game approach to free boundary problems of anisotropic forced mean curvature flow equations

Tue, 01 Apr 2025 00:00:00 GMT

Takuya Sato — We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was initially studied as the first-order Hamilton-Jacobi-Isaacs equations arising in pursuit-evasion differential games and applied to the models of first-order front propagation in Soravia (1994). In this paper, we consider an extension of these free boundary problems to the second-order equations and give a deterministic game representation based on a discrete approximation scheme in Kohn and Serfaty (2006). Furthermore, we prove the comparison principle for our free boundary problems by using the framework of time-discrete games.

Diffusion-aggregation equations and volume-preserving mean curvature flows

Tue, 25 Mar 2025 00:00:00 GMT

Jiwoong Jang, Antoine Mellet — The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation. It describes the aggregation of some organisms via chemotaxis, limited by some nonlinear diffusion. It is known that for some choice of this nonlinear diffusion, the PKS model asymptotically leads to phase separation and mean-curvature driven free boundary problems. In this paper, we focus on the Elliptic-Parabolic PKS model and we obtain the first unconditional convergence result in dimension 2 and 3 towards the volume preserving mean-curvature flow. This work builds up on previous results that were obtained under the assumption that phase separation does not cause energy loss in the limit. In order to avoid this assumption, we rely on Brakke type formulation of the mean-curvature flow and a reinterpretation of the problem as an Allen-Cahn equation with a nonlocal forcing term.

Neckpinch — Volume-preserving & constrained flows