Paper · Nov 11, 2018

Inverse Mean Curvature Flow with Singularities

Beomjun Choi, Pei-Ken Hung

math.DGWider flowsarXiv:1811.04497v2
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Abstract

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and each singularity is removed when the evolving cone becomes flat. As a result, we derive the exact waiting time for a weak solution to be a smooth solution. In particular, a necessary and sufficient condition for an existence of smooth classical solution is given.

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Submitted
Nov 11, 2018
Last revised
Feb 26, 2019 (v2)
Subjects
math.DG

Comments: 11 page, 1 figure, non-compact case included

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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of non-compact hypersurfaces by inverse mean curvature

    Beomjun Choi and Panagiota DaskalopoulosWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Same author Beomjun Choi · Shared terms “tangent cone”, “convex hypersurfaces”

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

    Brian HarvieWider flows

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

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

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  4. Inverse mean curvature flow with outer obstacle

    Kai XuWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows

  5. Global weak solutions for the inverse mean curvature flow in the Heisenberg group

    Adriano Pisante and Eugenio VecchiWider flows

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

  6. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows

  7. 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 “Lipschitz”

  8. Infinity-harmonic functions and inverse mean curvature flow clusters

    Kai XuWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows

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.