Paper · Sep 15, 2019

The Minkowski inequality in de Sitter space

Julian Scheuer

math.DGWider flowsarXiv:1909.06837
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Abstract

The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this paper we employ a locally constrained inverse mean curvature flow to prove a properly defined analogue in the Lorentzian de Sitter space.

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Submitted
Sep 15, 2019
Subjects
math.DG, math.AP
Journal
Pacific J. Math. 314 (2021) 425-449
DOI
10.2140/pjm.2021.314.425

Comments: 21 pages

Mean curvature flowOther curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsGeometric inequalitiesPhysics

Its neighbourhood

Connected papers

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 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)2Scheuer 2017 Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball (Julian Scheuer, Guofang Wang and Chao Xia, 2018)7Scheuer 2018 Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces (Ben Lambert and Julian Scheuer, 2019)6Lambert 2019 Minkowski inequalities and constrained inverse curvature flows in warped spaces (Julian Scheuer, 2020)1Scheuer 2020 The quermassintegral preserving mean curvature flow in the sphere (Esther Cabezas-Rivas and Julian Scheuer, 2022)5Cabezas-Riva 2022 New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms (Jie Wu, 2025)8Wu 2025 A locally constrained inverse Hessian quotient flow in de Sitter space (Kuicheng Ma, 2025)3Ma 2025 Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space (Kuicheng Ma, 2025)4Ma 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Minkowski inequalities and constrained inverse curvature flows in warped spaces

    Julian ScheuerWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Inverse mean curvature flow · Same author Julian Scheuer · Shared terms “Minkowski inequality”, “locally constrained”

  2. Locally constrained inverse curvature flows

    Julian Scheuer and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Inverse mean curvature flow · Same author Julian Scheuer · Shared terms “locally constrained”, “Sitter”, “Minkowski”

  3. A locally constrained inverse Hessian quotient flow in de Sitter space

    Kuicheng MaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Inverse mean curvature flow · Shared terms “Sitter space”, “locally constrained”

  4. Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

    Kuicheng MaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Inverse mean curvature flow · Shared terms “Sitter space”, “locally constrained”

  5. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Physics · Same author Julian Scheuer · Shared terms “Sitter space”

  6. Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces

    Ben Lambert and Julian ScheuerWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Physics · Same author Julian Scheuer · Shared terms “locally constrained”

  7. Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

    Julian Scheuer, Guofang Wang and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Same author Julian Scheuer · Shared terms “constrained inverse”, “locally constrained”, “Minkowski”

  8. New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

    Jie WuWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Inverse mean curvature flow · Shared terms “Minkowski inequality”, “locally constrained”

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986)

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.