Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

TL;DR

This paper studies a volume-preserving curvature flow for hypersurfaces with capillary boundary, showing that convex initial shapes evolve smoothly into spherical caps over time.

Contribution

It introduces a new volume-preserving curvature flow with non-local terms for hypersurfaces with capillary boundary and proves long-term convergence to spherical caps.

Findings

Flow exists for all time for convex initial hypersurfaces.

Flow smoothly converges to a spherical cap as time approaches infinity.

The flow preserves volume or area during evolution.

Abstract

In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate that for any convex initial hypersurface with a capillary boundary, the flow exists for all time and smoothly converges to a spherical cap as $t \to +\infty$