Paper · Oct 20, 2020

The Limit of the Inverse Mean Curvature Flow on a Torus

Brian Harvie

math.DGWider flowsarXiv:2010.10495v4
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Abstract

For an rotationally symmetric embedded torus , evolved by Inverse Mean Curvature Flow, we show that the total curvature remains bounded up to the singular time . We then show convergence of the to a rotationally symmetric embedded torus as without rescaling. Later, we observe a scale-invariant energy estimate on any embedded solution of the flow in that may be useful in ruling out curvature blowup near singularities in general.

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Submitted
Oct 20, 2020
Last revised
Sep 3, 2021 (v4)
Subjects
math.DG
Journal
Proc. Amer. Math. Soc. 150 (2022), 3049-3061
DOI
10.1090/proc/15812

Comments: 15 pages, 2 figures, some minor typos corrected. Final version to appear in Proceedings of the American Mathematical Society

Mean curvature flowOther curvature flowsInverse mean curvature flowSingularitiesRotational symmetry & warped products

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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 Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends (Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo Lee, 2016)8Drugan 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)7Mullins 2016 Existence of self-similar solution of the inverse mean curvature flow (K. M. Hui, 2018)3Hui 2018 Existence of hypercylinder expanders of the inverse mean curvature flow (K. M. Hui, 2018)6Hui 2018 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)4Chang 2018 Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces (Brian Harvie, 2019)2Harvie 2019 Another proof of the existence of homothetic solitons of the inverse mean curvature flow (Shu-Yu Hsu, 2020)5Hsu 2020 Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces (Brian Harvie, 2020)1Harvie 2020
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 of Rotationally Symmetric Hypersurfaces

    Brian HarvieWider flows

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

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

    Brian HarvieWider flows

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

  3. Existence of self-similar solution of the inverse mean curvature flow

    K. M. HuiWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows

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

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  5. Another proof of the existence of homothetic solitons of the inverse mean curvature flow

    Shu-Yu HsuWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows

  6. Existence of hypercylinder expanders of the inverse mean curvature flow

    K. M. Hui

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows

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

    Thomas MullinsWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows

  8. Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends

    Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo LeeWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows · Shared terms “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) 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. 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.

  2. 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.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.