Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

Brian Allen

Paper · Jul 23, 2018

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

Brian Allen

math.DGWider flowsarXiv:1807.08822v2

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Abstract

We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_{T}^{i} \subset M_{i}^{3}$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if $\partial U_{T}^{i} = Σ_{0}^{i} \cup Σ_{T}^{i}$ , $m_{H} (Σ_{T}^{i}) \to 0$ and extra technical conditions are satisfied we show that $U_{T}^{i}$ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.

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Submitted: Jul 23, 2018
Last revised: Dec 14, 2021 (v2)
Subjects: math.DG, gr-qc, math.MG

Comments: Comments welcome, 51 pages, 10 figures. v2: Adjustments made throughout for final version to appear in Communications in Analysis and Geometry

Mean curvature flow Other curvature flows Inverse mean curvature flow Scalar curvature Stability & uniqueness 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
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] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[2] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[3] 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.

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

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