Paper · Nov 15, 2018

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

Yuguang Shi, Jintian Zhu

math.DGWider flowsarXiv:1811.06158v5
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Abstract

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected -smooth surface as initial data in asymptotically ADS-Schwarzschild manifolds with positive mass is bigger than or equal to the total mass, which is completely different from the situation in asymptotically flat case.

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Submitted
Nov 15, 2018
Last revised
Mar 13, 2021 (v5)
Subjects
math.DG

Comments: 18 pages, 1 figure, accepted for publication in SCIENCE CHINA Mathematics, all comments are welcome

Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsWeak & singular flowsBoundaries & non-compactPhysics

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    Liam Mazurowski and Jintian ZhuWider flows

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Same author Jintian Zhu · Shared terms “asymptotically hyperbolic”, “hyperbolic manifolds”, “asymptotically flat”

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

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

  3. On the evolution of hypersurfaces along their inverse spacetime mean curvature

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    Topics Inverse mean curvature flow, Physics, Weak & singular flows · Shared terms “asymptotically flat”, “initial data”

  4. Hawking Mass Monotonicity for Initial Data Sets

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    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Hawking mass”, “initial data”

  5. Asymptotically Hyperbolic 3-Metric with Ricci flow foliation

    Hyun Chul Jang

    Topics Physics, Boundaries & non-compact · Shared terms “asymptotically hyperbolic”, “total mass”, “asymptotically flat”

  6. Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold

    Li Chen and Jing MaoWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “AdS Schwarzschild”, “Schwarzschild manifold”, “star shaped”

  7. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Shared terms “Schwarzschild manifold”, “asymptotically flat”, “star shaped”

  8. Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

    Mattia Fogagnolo, Giorgio Gatti and Alessandra PludaWider flows

    Topics Inverse mean curvature flow, Physics, 3-manifolds · Shared terms “Hawking mass”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.