Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds

TL;DR

This paper constructs an example of inverse mean curvature flow on asymptotically hyperbolic 3-manifolds that fails to converge as needed for proving Penrose-type inequalities, highlighting differences from the flat case.

Contribution

It demonstrates the failure of convergence in inverse mean curvature flow on asymptotically hyperbolic manifolds, contrasting with known results in asymptotically flat geometries.

Findings

Constructed a non-convergent inverse mean curvature flow solution

Identified key differences between hyperbolic and flat cases

Derived a new inequality assuming Penrose inequality holds

Abstract

We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi--Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$.