Long Time Existence of Non-Compact Inverse Mean Curvature Flow in Hyperbolic Space

TL;DR

This paper proves the long-term existence of inverse mean curvature flow for certain non-compact hypersurfaces in hyperbolic space, providing key estimates and tools for the analysis.

Contribution

It establishes long time existence results for IMCF of non-compact hypersurfaces in hyperbolic space, introducing new estimates and maximum principles.

Findings

Long time existence of IMCF for bounded graphs over horospheres.

Development of cutoff functions and maximum principles at infinity.

Derivation of local and global estimates for the flow.

Abstract

We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in $\mathbb{H}^{n+1}$ and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition, we develop a useful family of cutoff functions for IMCF as well as a non-compact ODE maximum principle at infinity which are integral tools used throughout the document.