Paper · Sep 3, 2019

Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

Brian Harvie

math.DGWider flowsarXiv:1909.01328v5
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Abstract

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection within a prescribed time interval depending only on initial data. Finally, we establish the existence of either finite-time singularities or intersections for certain topological spheres under IMCF.

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Submitted
Sep 3, 2019
Last revised
Jun 8, 2021 (v5)
Subjects
math.DG
Journal
Math. Res. Lett. 29 (2023), 1065-1086
DOI
10.4310/MRL.2022.v29.n4.a7

Comments: 22 pages, 2 figures. Final version to appear in Mathematical Research Letters. Added clarification in Section 3 on how one relates the global variational solution of IMCF to the variational solution induced by the classical flow. The most recent update was to fix some typos and update contact info in the document, but no further edits were made

Mean curvature flowOther curvature flowsInverse mean curvature flowSingularitiesWeak & singular flows

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 Inverse mean curvature flow in complex hyperbolic space (Giuseppe Pipoli, 2016)7Pipoli 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)4Mullins 2016 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)6Chang 2018 Inverse Mean Curvature Flow with Singularities (Beomjun Choi and Pei-Ken Hung, 2018)1Choi 2018 Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces (Brian Harvie, 2020)3Harvie 2020 The Limit of the Inverse Mean Curvature Flow on a Torus (Brian Harvie, 2020)5Harvie 2020 On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space (Brian Harvie, 2024)2Harvie 2024 Local regularity and p-harmonic approximation for weak inverse mean curvature flow (Wei-Chun Chen and Chun-Kai Lien, 2026)8Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse Mean Curvature Flow with Singularities

    Beomjun Choi and Pei-Ken HungWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “waiting time”

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

    Brian HarvieWider flows

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

  3. Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

    Brian HarvieWider flows

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

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

    Thomas MullinsWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “shaped surface”, “star shaped”

  5. The Limit of the Inverse Mean Curvature Flow on a Torus

    Brian HarvieWider flows

    Topics Inverse mean curvature flow, Other curvature flows, Singularities · Same author Brian Harvie

  6. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  7. Inverse mean curvature flow in complex hyperbolic space

    Giuseppe PipoliWider flows

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

  8. Local regularity and p-harmonic approximation for weak inverse mean curvature flow

    Wei-Chun Chen and Chun-Kai LienWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “variational weak”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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

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

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.