ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

TL;DR

This paper extends the ODE maximum principle to non-compact hypersurfaces and applies it to study the long-term behavior of inverse mean curvature flow in hyperbolic space, showing convergence to horospheres.

Contribution

It introduces a new maximum principle at infinity for non-compact hypersurfaces and applies it to analyze IMCF in hyperbolic space, demonstrating long-term existence and convergence.

Findings

Flow exists for a long time on bounded graphs over horospheres.

IMCF solutions asymptotically approach horospheres.

Extended maximum principle aids in analyzing non-compact hypersurfaces.

Abstract

In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, 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 as well as asymptotic convergence to horospheres.