Paper · Apr 14, 2017

Capacity inequalities and rigidity of cornered/conical manifolds

C. Tiarlos Cruz

math.DGWider flowsarXiv:1704.04306v2
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Abstract

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.

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Submitted
Apr 14, 2017
Last revised
Aug 29, 2018 (v2)
Subjects
math.DG, math.AP

Comments: Revised version. 18 pages; comments are welcome

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

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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 An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere (Frederico Girão and Neilha M. Pinheiro, 2016)1Girão 2016 Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under L² Metric Convergence (Brian Allen, 2017)6Allen 2017 A Penrose type inequaltiy for graphs over Reissner-Nordström-anti-deSitter manifold (Daguang Chen, Haizhong Li and Tailong Zhou, 2017)3Chen 2017 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)5Andrews 2019 The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary (Thomas Koerber, 2019)2Koerber 2019 The asymptotic behaviour of p-capacitary potentials in Asymptotically Conical manifolds (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)4Benatti 2022 Monotone Quantities for p-Harmonic functions and the Sharp p-Penrose inequality (Liam Mazurowski and Xuan Yao, 2023)7Mazurowski 2023 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)8Pan 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere

    Frederico Girão and Neilha M. PinheiroWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”, “type inequality”

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

    Thomas KoerberWider flows

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

  3. A Penrose type inequaltiy for graphs over Reissner-Nordström-anti-deSitter manifold

    Daguang Chen, Haizhong Li and Tailong ZhouWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Penrose type”, “Alexandrov Fenchel”, “type inequality”

  4. The asymptotic behaviour of -capacitary potentials in Asymptotically Conical manifolds

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Shared terms “conical manifolds”

  5. Harmonic mean curvature flow and geometric inequalities

    Ben Andrews, Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”, “type inequality”

  6. 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, Boundaries & non-compact · Shared terms “Penrose”, “flat manifolds”, “asymptotically flat”

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

    Liam Mazurowski and Xuan YaoWider flows

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

  8. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Boundaries & non-compact · Shared terms “type inequality”, “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) 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.