Sobolev stability of the PMT and RPI using IMCF

Abstract

We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M_i^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time $t\in [0,T]$. In particular, we consider a sequence of regions of manifolds $U_T^i\subset M_i^3$, foliated by a IMCF, ${\Sigma }_t$, such that 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 in $W^{1,2}$ to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. 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 in $W^{1,2}$ to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de~Sitter Schwarzschild metric.

arXiv comment37 pages, comments welcome. Some definitions and important estimates are repeated from arXiv:1705.00591, arXiv:1707.09388 for the convenience of the reader. The additional $L^{\infty }$ estimates of the first version have been removed due to issues with the proof. Final version to appear in General Relativity and Gravitation

References in corpus (15)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality
• Penrose type inequalities for asymptotically hyperbolic graphs
• The inverse mean curvature flow in warped cylinders of non-positive radial curvature
• Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds
• A Level Set Analysis of the Witten Spinor with Applications to Curvature Estimates
• Inverse curvature flows in Riemannian warped products
• Contrasting Various Notions of Convergence in Geometric Analysis
• 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
• On the stability of the positive mass theorem for asymptotically hyperbolic graphs
• Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow
• Asymptotically hyperbolic manifolds with small mass
• Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

Cited by in corpus (3)

• Properties of the Null Distance and Spacetime Convergence
• Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass
• Stability of the positive mass theorem under Ricci curvature lower bounds