A constrained mean curvature flow and Alexandrov-Fenchel inequalities

TL;DR

This paper introduces a constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary, proving long-term existence, convergence to spherical caps, and deriving new Alexandrov-Fenchel inequalities for convex hypersurfaces.

Contribution

It presents a novel constrained curvature flow in the half-space setting and establishes new geometric inequalities for hypersurfaces with capillary boundary.

Findings

Flow exists long-term and converges to spherical caps.

Capillary quermassintegrals are monotonic along the flow.

New Alexandrov-Fenchel inequalities are established for convex hypersurfaces.

Abstract

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.