A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane

TL;DR

This paper introduces a new volume-preserving mean curvature flow for capillary hypersurfaces in hyperbolic space, proving its long-term existence, convergence, and identifying energy minimizers among such hypersurfaces.

Contribution

It establishes a Minkowski type formula and demonstrates the flow's convergence to a totally umbilical cap, revealing new geometric insights in hyperbolic space.

Findings

Flow exists for all time and converges uniformly to a totally umbilical cap.

A new Minkowski type formula for capillary hypersurfaces is proved.

Totally umbilical caps are energy minimizers for fixed volume.

Abstract

We prove a new Minkowski type formula for capillary hypersurfaces supported on totally geodesic hyperplanes in hyperbolic space. It leads to a volume-preserving flow starting from a star-shaped initial hypersurface. We prove the long-time existence of the flow and its uniform convergence to a $\theta$-totally umbilical cap. Additionally, we establish that a $\theta$-totally umbilical cap is an energy minimizer for a given enclosed volume.