Paper · Jan 18, 2017

On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space

Yong Wei

math.DGWider flowsarXiv:1701.04964v2
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Abstract

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.

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Submitted
Jan 18, 2017
Last revised
Mar 1, 2018 (v2)
Subjects
math.DG, math.AP
Journal
Calc.Var.Partial Differ. Equ., (2018) 57: 46
DOI
10.1007/s00526-018-1342-x

Comments: minor revision, accepted by Calculus of Variations and PDE

Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsGeometric inequalities

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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)5Girão 2016 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)7Andrews 2019 On an inverse curvature flow in two-dimensional space forms (Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al., 2021)8Kwong 2021 On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space (Brian Harvie, 2024)3Harvie 2024 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)1Pan 2024 Upper bounds for the Alexandrov-Fenchel deficit via integral formulas (Kwok-Kun Kwong and Yong Wei, 2025)4Kwong 2025 Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces (Fang Hong, 2026)6Hong 2026 Weinstock inequalities for outward-minimizing domains (Chaoqun Gao, Yong Wei and Rong Zhou, 2026)2Gao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Weak & singular flows · Shared terms “Minkowski type”, “type inequality”, “minimizing hypersurfaces”

  2. Weinstock inequalities for outward-minimizing domains

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Yong Wei · Shared terms “outward minimizing”

  3. On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space

    Brian HarvieWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Weak & singular flows · Shared terms “Minkowski type”, “type inequality”, “minimizing”

  4. Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

    Kwok-Kun Kwong and Yong WeiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Yong Wei · Shared terms “Minkowski”

  5. 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 “type inequality”

  6. Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

    Fang HongWider flows

    Topics Geometric inequalities · Shared terms “Minkowski type”, “sharp Minkowski”, “type inequality”

  7. 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 “type inequality”

  8. On an inverse curvature flow in two-dimensional space forms

    Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al.Wider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Yong Wei

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.