Paper · Jul 28, 2017

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

Brian Allen

math.DGWider flowsarXiv:1707.09388v3
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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 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 , foliated by a smooth solution to IMCF which is uniformly controlled, and if and then converges to a topological annulus portion of hyperbolic space with respect to metric convergence. If instead and then we show that converges to a topological annulus portion of the Anti-deSitter Schwarzschild metric with respect to metric convergence.

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Submitted
Jul 28, 2017
Last revised
Apr 13, 2018 (v3)
Subjects
math.DG, gr-qc
Journal
Journal of Mathematical Physics 59, 082501 (2018)
DOI
10.1063/1.5035275

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

Mean curvature flowOther curvature flowsInverse mean curvature flowStability & uniquenessBoundaries & non-compactGeometric inequalitiesPhysics

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under L² Metric Convergence (Brian Allen, 2017)1Allen 2017 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)4Allen 2017 Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem (Brian Allen, 2018)3Allen 2018 Sobolev stability of the PMT and RPI using IMCF (Brian Allen, 2018)2Allen 2018 Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 (Yuguang Shi and Jintian Zhu, 2018)7Shi 2018 The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary (Thomas Koerber, 2019)5Koerber 2019 On the Isoperimetric Riemannian Penrose Inequality (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)6Benatti 2022 The Riemannian Penrose Inequality with Matter Density (Hubert Bray and Yiyue Zhang, 2024)8Bray 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under Metric Convergence

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Brian Allen · Shared terms “metric convergence”, “Riemannian Penrose”, “Penrose inequality”

  2. Sobolev stability of the PMT and RPI using IMCF

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Brian Allen · Shared terms “annulus portion”, “topological annulus”, “foliated”

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

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Same author Brian Allen · Shared terms “mass theorem”, “positive mass”, “uniformly controlled”

  4. Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Brian Allen · Shared terms “Riemannian Penrose”, “IMCF”, “Penrose inequality”

  5. The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary

    Thomas KoerberWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”, “Riemannian Penrose”

  6. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Riemannian Penrose”, “Penrose inequality”

  7. Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension

    Yuguang Shi and Jintian ZhuWider flows

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “hyperbolic manifolds”, “asymptotically hyperbolic”, “positive mass”

  8. The Riemannian Penrose Inequality with Matter Density

    Hubert Bray and Yiyue ZhangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Riemannian Penrose”, “Penrose inequality”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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)

In the guidebook

  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.