Paper · Aug 21, 2017

Locally constrained inverse curvature flows

Julian Scheuer, Chao Xia

math.DGWider flowsarXiv:1708.06125v2
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Abstract

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski type inequality in the anti-de-Sitter Schwarzschild manifolds and a weighted isoperimetric type inequality in the hyperbolic space.

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Submitted
Aug 21, 2017
Last revised
Oct 26, 2017 (v2)
Subjects
math.DG, math.AP
Journal
Trans. Am. Math. Soc. 372, no. 10, p. 6771-6803 (2019)
DOI
10.1090/tran/7949

Comments: The proof of Proposition 7.6 has been minor revised

Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergenceRotational symmetry & warped productsGeometric inequalitiesPhysics

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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 Inverse curvature flow in anti-de Sitter-Schwarzschild manifold (Siyuan Lu, 2016)7Lu 2016 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)6Scheuer 2017 Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball (Julian Scheuer, Guofang Wang and Chao Xia, 2018)2Scheuer 2018 The Minkowski inequality in de Sitter space (Julian Scheuer, 2019)3Scheuer 2019 Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces (Ben Lambert and Julian Scheuer, 2019)8Lambert 2019 Minkowski inequalities and constrained inverse curvature flows in warped spaces (Julian Scheuer, 2020)1Scheuer 2020 A locally constrained inverse Hessian quotient flow in de Sitter space (Kuicheng Ma, 2025)5Ma 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 “locally constrained”, “coordinate slice”, “radial coordinate”

  2. 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 authors Julian Scheuer, Chao Xia · Shared terms “constrained inverse”, “type inequality”, “locally constrained”

  3. 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 “locally constrained”, “Sitter”, “Minkowski”

  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 “locally constrained”, “type inequality”, “Sitter”

  5. 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”, “Sitter”

  6. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence · Same author Julian Scheuer

  7. Inverse curvature flow in anti-de Sitter-Schwarzschild manifold

    Siyuan LuWider flows

    Topics Inverse mean curvature flow, Physics, Long-time existence & convergence · Shared terms “Schwarzschild manifold”, “Sitter Schwarzschild”

  8. 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”, “isoperimetric”

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.

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)