Volume-preserving & constrained flows

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 $$⟨X+AX,\nu ⟩>0$$ for some fixed skew-symmetric linear map $A$, where $X$ is the position vector and $\nu$ is the outward unit normal. The vector field $X+AX$ is conformal Killing, with dilation component $X$ and rotational component $AX$; 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.

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 ${\kappa }_{\sigma }^{\alpha }$, where $0<\alpha \le 1$. Here ${\kappa }_{\sigma }$ denotes the anisotropic curvature associated with a prescribed anisotropy function $\sigma$. 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 $V={\kappa }_{\sigma }^{\alpha }$. 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.

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 $(1\le l<k\le n)$. Using this flow, we establish a full family of the quermassintegral inequalities for horo-convex capillary hypersurfaces in a horoball with contact angle $\theta \in (0,\pi /2]$. 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.

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.

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 $\alpha$-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any $\alpha >0$ 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.

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 $u_x/u$ converges to one away from the symmetry axis. The normalized velocity $u_t/u$ 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.

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 ${\theta }_{\pm }$, the angle $\theta$ between the skew lines, and the initial data ${\gamma }_0$, 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.

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.

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 $L^2$ distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on $\phi$ 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 $\phi$-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 $\phi$-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.

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.

In this paper, we study the area-preserving and length-preserving ${\kappa }^{\alpha }$-type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane ${\mathbb{H}}^2$ for $\alpha <0$ 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.

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.

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 $\theta \in (0,\pi )$ 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.

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.

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 $$v=-H-{\partial }_{\nu }\log |\nabla u|+f(u)/|\nabla u|,$$ satisfied by level surfaces of any solution to the nonlinear parabolic equation $${\partial }_{t}u=\Delta u-f(u).$$ 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 \left\{\begin{alignedat}{2} \partial_tu&=Δu\quad&&\text{in}\quad\{|u|<1\} |\nabla u|&=1/ε\quad&&\text{on}\quad\partial\{|u|<1\}, \end{alignedat} \right. and confirm that under reasonable assumptions, the $C^{\alpha }$ norm of the forcing term ${\partial }_{\nu }\log |\nabla u|$ converges to zero at an algebraic rate as $\epsilon \to 0$, uniformly in time. This implies that the parabolic free boundary Allen–Cahn equation converges to the mean curvature flow, uniformly (in $\epsilon$ and in time) in the $C^{2,\alpha }$ sense.

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 $R_{\Lambda }({\Omega }_t)$. For convex domains we show that $R_{\Lambda }$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_{\Lambda }$ and the mean curvature $H$, established in all dimensions: in $d=2$ via a near-disk Fourier analysis, and in $d\ge 3$ 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.

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.

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 $W\in (-\infty ,{\bar{W}}_0)\cup \{0\}\cup (W_0,+\infty )$, where ${\bar{W}}_0<0$ and $W_0>0$ 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 $W\ge 0$, wherea it is a geodesic of nonvanishing curvature when $W$ is a negative constant. Specially, in the first Heisenberg group $\mathbb{M}(0)$, 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 $\mathbb{M}(0)$ and provide a new proof of the fact that the total curvature of $\gamma \subset \mathbb{M}(0)$ with strictly positive curvature equals $2\pi$.

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 $L^2$-estimate of multipliers. The difficulty is handled by conducting blowup analysis at a point in contact to a spherical cap with sharp angle.