Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

TL;DR

This paper introduces a new modified mean curvature flow in hyperbolic space, establishing existence, uniqueness, and convergence results for star-shaped hypersurfaces with fixed boundary at infinity, and recovers known results on constant mean curvature hypersurfaces.

Contribution

It defines a novel MMCF in hyperbolic space linked to a specific energy functional and proves fundamental properties, extending the understanding of hypersurfaces with prescribed asymptotic boundaries.

Findings

Existence and uniqueness of the MMCF for star-shaped hypersurfaces.

Convergence of the flow to hypersurfaces with constant mean curvature.

Recovery of classical results on hypersurfaces with prescribed boundary at infinity.

Abstract

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, which was first shown by Guan and Spruck.