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

TL;DR

This paper introduces a new geometric flow for capillary hypersurfaces in space forms, demonstrating its long-term existence, convergence to spherical caps, and its application to proving a capillary isoperimetric inequality.

Contribution

The paper develops a novel constrained mean curvature flow that preserves volume and decreases energy, providing a new proof of the capillary isoperimetric inequality.

Findings

Flow exists for all time and converges to spherical caps.

Flow preserves volume and decreases total energy.

Provides a flow-based proof of the capillary isoperimetric inequality.

Abstract

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.