Paper · Aug 17, 2020

Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

Brian Harvie

math.DGWider flowsarXiv:2008.07490v4
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Abstract

We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as . Our approach is based on a localized version of the parabolic maximum principle. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in . The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in .

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Submitted
Aug 17, 2020
Last revised
Nov 21, 2022 (v4)
Subjects
math.DG, math.AP
Journal
Calc. Var. PDE. 62 (2023), No. 125
DOI
10.1007/s00526-023-02469-8

Comments: 40 pages, 7 figures. This version includes more exposition on rotationally symmetric immersions, some additional applications, and a simpler example of a non-star-shaped admissible initial surface

Mean curvature flowOther curvature flowsInverse mean curvature flowLong-time existence & convergenceRotational symmetry & warped productsGeometric inequalitiesMinimal & CMC surfaces

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 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)5Mullins 2016 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)4Hu 2016 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)7Chen 2017 Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries (Kunbo Wang, 2017)8Wang 2017 Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces (Brian Harvie, 2019)3Harvie 2019 The Limit of the Inverse Mean Curvature Flow on a Torus (Brian Harvie, 2020)1Harvie 2020 On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space (Brian Harvie, 2024)2Harvie 2024 On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes (Brian Harvie and Ye-Kai Wang, 2025)6Harvie 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Limit of the Inverse Mean Curvature Flow on a Torus

    Brian HarvieWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows · Same author Brian Harvie · Shared terms “rotationally symmetric”

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

    Brian HarvieWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Brian Harvie · Shared terms “star shaped”, “Minkowski inequality”, “IMCF”

  3. Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

    Brian HarvieWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Brian Harvie · Shared terms “star shaped”, “IMCF”

  4. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “star shaped”, “mean convex”

  5. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “star shaped”

  6. On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes

    Brian Harvie and Ye-Kai WangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Brian Harvie · Shared terms “IMCF”, “Minkowski inequality”

  7. Inverse mean curvature flow inside a cone in warped products

    Li Chen, Jing Mao, Ni Xiang et al.Wider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “star shaped”, “round sphere”

  8. Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

    Kunbo WangWider flows

    Topics Minimal & CMC surfaces, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “symmetric hypersurfaces”, “rotationally symmetric”

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) 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)

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.