Paper · Sep 29, 2019

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

Thomas Koerber

math.DGWider flowsarXiv:1909.13283v2
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Abstract

In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation scheme for the weak free boundary inverse mean curvature flow, introduced by Marquardt in 2012, and establish the monotonicity of a free boundary version of the Hawking mass. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in and thus sheds some light on a conjecture made by Huisken.

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Submitted
Sep 29, 2019
Last revised
Jan 14, 2020 (v2)
Subjects
math.DG

Comments: 42 pages, some minor changes and corrected typos, all comments welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compactFree boundary & capillaryGeometric inequalitiesPhysics

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  1. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

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

  2. 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 · Shared terms “Penrose inequality”, “Riemannian Penrose”, “asymptotically flat”

  3. The Penrose inequality with charge for 2-convex initial data sets

    Tuan DolmenWider flows

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

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

    Brian AllenWider flows

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

  5. The Riemannian Penrose Inequality with Matter Density

    Hubert Bray and Yiyue ZhangWider flows

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

  6. Capacity inequalities and rigidity of cornered/conical manifolds

    C. Tiarlos CruzWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Boundaries & non-compact · Shared terms “asymptotically flat”, “flat manifolds”, “Penrose”

  7. Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality

    Liam Mazurowski and Xuan YaoWider flows

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

  8. The -inverse mean curvature flow and the generalized Penrose conjecture

    Conghan DongWider flows

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “Penrose”, “asymptotically flat”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.