A constrained mean curvature type flow for capillary boundary hypersurfaces in space forms

In this paper, we introduce a new constrained mean curvature type flow for capillary boundary hypersurfaces in space forms. We show the flow exists for all time and converges globally to a spherical cap. Moreover, the flow preserves the volume of the bounded domain enclosed by the hypersurface and decreases the total energy. As a by-product, we give a flow proof of the capillary isoperimetric inequality for the starshaped capillary boundary hypersurfaces in space forms.Comment: 23page

A constrained mean curvature flow and Alexandrov-Fenchel inequalities

In this article, we study a locally constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary in the half-space. We prove its long-time existence and the global convergence to a spherical cap. Furthermore, the capillary quermassintegrals defined in \cite{WWX2022} evolve monotonically along the flow, and hence we establish a class of new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space.Comment: Final version, to appear in Int. Math. Res. Not. IMR

The relative isoperimetric inequality for minimal submanifolds in the Euclidean space

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds in $\mathbb{R}^{n+m}$. We first provide, following Cabr\'e \cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved in Choe-Ghomi-Ritor\'e \cite{CGR07}, by generalizing ideas of restricted normal cones given in \cite{CGR06}. Then we prove a relative isoperimetric inequalities for minimal submanifolds in $\mathbb{R}^{n+m}$, which is optimal when the codimension $m\le 2$. In other words we obtain a relative version of isoperimetric inequalities for minimal submanifolds proved recently by Brendle \cite{Brendle2019}. When the codimension $m\le 2$, our result gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6. As another application we prove an optimal logarithmic Sobolev inequality for free boundary submanifolds in the Euclidean space following a trick of Brendle in \cite{Brendle2019b}.Comment: 18 page

Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov-Fenchel inequalities. In order to prove these results, we construct a new locally constrained curvature flow and prove that the flow converges globally to a spherical cap.Comment: 25 pages, 1 figure. Comments are welcom