Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by 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 $m_H({\Sigma }_T^i)\to 0$ then $U_T^i$ converges to a topological annulus portion of hyperbolic space with respect to $L^2$ metric convergence. If instead $m_H({\Sigma }_T^i)-m_H({\Sigma }_0^i)\to 0$ and $m_H({\Sigma }_T^i)\to m>0$ 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.

29 pages. Typos corrected. Preliminaries, estimates, and some proofs are repeated from arXiv:1705.00591 for the convenience of the reader

References in corpus (5)

• Inverse curvature flows in Riemannian warped products
• Inverse mean curvature flows in warped product manifolds
• Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry
• Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^2$ Metric Convergence
• ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

Cited by in corpus (6)

• Properties of the Null Distance and Spacetime Convergence
• Contrasting Various Notions of Convergence in Geometric Analysis
• Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^2$ Metric Convergence
• On the stability of the positive mass theorem for asymptotically hyperbolic graphs
• Sobolev stability of the PMT and RPI using IMCF
• Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds