A fully nonlinear locally constrained curvature flow for capillary hypersurface

TL;DR

This paper introduces a new fully nonlinear curvature flow for convex capillary hypersurfaces in half-space, demonstrating its long-term existence, convexity preservation, and convergence to a spherical cap, with implications for geometric inequalities.

Contribution

It develops a fully nonlinear, locally constrained curvature flow for convex capillary hypersurfaces, extending previous results and deriving new geometric inequalities.

Findings

Flow preserves convexity and exists for all time.

Flow converges smoothly to a spherical cap.

Monotonic evolution of a high-order capillary isoperimetric ratio.

Abstract

In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in \cite{MWW}. As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov-Fenchel inequalities.