A mean curvature type flow with capillary boundary in a horoball in hyperbolic space

TL;DR

This paper introduces a mean curvature flow with capillary boundary conditions in hyperbolic space, demonstrating long-term existence and convergence to a special hypersurface, and solving an isoperimetric problem.

Contribution

It develops a novel mean curvature flow with capillary boundary in hyperbolic space and proves its convergence, addressing an open isoperimetric problem in this setting.

Findings

Flow preserves volume and decreases energy functional

Flow exists for all time and converges to a truncated umbilical hypersurface

Solves an isoperimetric problem with capillary boundary in a horoball

Abstract

In this paper, we study a mean curvature type flow with capillary boundary in a horoball in hyperbolic space. Our flow preserves the volume of the bounded domain enclosed by the hypersurface and monotonically decreases the energy functional. We show that it has the long time existence and converges to a truncated umbilical hypersurface in hyperbolic space. As an application, we solve an isoperimetric type problem for hypersurfaces with capillary boundary in a horoball.