Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities

TL;DR

This paper introduces an anisotropic volume-preserving mean curvature flow for star-shaped capillary hypersurfaces, proving long-term existence, convergence to Wulff shapes, and establishing new inequalities in anisotropic convex geometry.

Contribution

It develops a novel anisotropic flow approach for capillary hypersurfaces, extending quermassintegrals and deriving new Alexandrov-Fenchel inequalities.

Findings

Flow exists long-term and converges smoothly to Wulff shapes.

Generalized quermassintegrals satisfy monotonicity along the flow.

Established new anisotropic capillary isoperimetric and Alexandrov-Fenchel inequalities.

Abstract

In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic $\omega_0$-capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all $t>0$ by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the $\omega_0$-capillary Wulff shape is a $\theta$-capillary hypersurface with constant contact angle $\theta$, the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two $\theta$-capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the $\omega_0$-capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two $\theta$-capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry.