Sobolev stability of the PMT and RPI using IMCF

Brian Allen

Paper · Aug 23, 2018

Sobolev stability of the PMT and RPI using IMCF

Brian Allen

math.DGWider flowsarXiv:1808.07841v2

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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, $Σ_{t}$ , such that if $\partial U_{T}^{i} = Σ_{0}^{i} \cup Σ_{T}^{i}$ and $m_{H} (Σ_{T}^{i}) \to 0$ 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 $m_{H} (Σ_{T}^{i}) - m_{H} (Σ_{0}^{i}) \to 0$ and $m_{H} (Σ_{T}^{i}) \to m > 0$ 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.

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Submitted: Aug 23, 2018
Last revised: Apr 5, 2019 (v2)
Subjects: math.DG, gr-qc
Journal: General Relativity and Gravitation (2019) 51: 59
DOI: 10.1007/s10714-019-2542-1

Comments: 37 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

Mean curvature flow Other curvature flows Inverse mean curvature flow Stability & uniqueness Geometric inequalities Physics

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

[1] 6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Sobolev stability of the PMT and RPI using IMCF

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