Paper · May 22, 2020

Minkowski inequalities and constrained inverse curvature flows in warped spaces

Julian Scheuer

math.DGWider flowsarXiv:2005.11236
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Abstract

This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows we prove long time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases new Minkowski type inequalities are the consequence. In higher dimensions we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

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Submitted
May 22, 2020
Subjects
math.DG, math.AP, math.MG
Journal
Adv. Calc. Var. 15, no. 4, p. 735-748, (2022)
DOI
10.1515/acv-2020-0050

Comments: 20 pages

Mean curvature flowOther curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsSurfacesLong-time existence & convergenceGeometric 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 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)1Scheuer 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)3Scheuer 2017 Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball (Julian Scheuer, Guofang Wang and Chao Xia, 2018)5Scheuer 2018 The Minkowski inequality in de Sitter space (Julian Scheuer, 2019)2Scheuer 2019 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)8Pan 2024 New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms (Jie Wu, 2025)4Wu 2025 A locally constrained inverse Hessian quotient flow in de Sitter space (Kuicheng Ma, 2025)7Ma 2025 Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space (Kuicheng Ma, 2025)6Ma 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”, “coordinate slice”, “radial coordinate”

  2. The Minkowski inequality in de Sitter space

    Julian ScheuerWider flows

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

  3. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Julian Scheuer · Shared terms “Riemannian warped”

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

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

  5. 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”, “type inequality”, “Minkowski type”

  6. 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 “locally constrained”, “type inequality”

  7. 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 “locally constrained”

  8. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

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

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.