Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

TL;DR

This paper investigates the stability of the Positive Mass Theorem and Riemannian Penrose Inequality for asymptotically hyperbolic manifolds, showing convergence to hyperbolic space or Anti-deSitter Schwarzschild space under certain conditions using inverse mean curvature flow.

Contribution

It establishes conditions under which regions of asymptotically hyperbolic manifolds converge to model spaces, extending stability results for the PMT and RPI.

Findings

Regions with vanishing Hawking mass converge to hyperbolic space.

Regions with Hawking mass approaching a positive constant converge to Anti-deSitter Schwarzschild space.

Provides a framework for stability analysis using inverse mean curvature flow.

Abstract

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically hyperbolic manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically hyperbolic manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i = \Sigma_0^i \cup \Sigma_T^i$ and $m_H(\Sigma_T^i) \rightarrow 0$ then $U_T^i$ converges to a topological annulus portion of hyperbolic space with respect to $L^2$ metric convergence. If instead $m_H(\Sigma_T^i)-m_H(\Sigma_0^i) \rightarrow 0$ and $m_H(\Sigma_T^i) \rightarrow m >0$ then we show that $U_T^i$ converges to a topological annulus portion of the Anti-deSitter Schwarzschild metric with respect to $L^2$ metric convergence.