Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

TL;DR

This paper studies a modified mean curvature flow in hyperbolic space, proving long-term existence for entire radial graphs and exploring its behavior in relation to the asymptotic Plateau problem.

Contribution

It establishes the existence and persistence of the modified mean curvature flow for entire locally Lipschitz radial graphs in hyperbolic space.

Findings

Flow exists for all time starting from such graphs.

Flow remains radially graphic throughout its evolution.

Convergence of the flow is not guaranteed in general.

Abstract

The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in $\mathbb{H}^{n+1}$. Similar to the usual mean curvature flow, the MMCF is the natural negative $L^2$-gradient flow of the area-volume functional $\mathcal{I}(\Sigma)=A(\Sigma)+\sigma V(\Sigma)$ associated to a hypersurface $\Sigma$. In this paper, we prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time. In general one cannot expect the convergence of the flow as it can be seen from the flow starting from a horosphere (whose asymptotic boundary is degenerate to a point).