Paper · Dec 1, 2023

Mean Curvature Flow from Conical Singularities

Otis Chodosh, J. M. Daniels-Holgate, Felix Schulze

math.DGWider flowsarXiv:2312.00759v2
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Abstract

We prove Ilmanen's resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface , , with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.

Comments: 22 pages, 3 figures. Final version, to appear in Invent. Math

Mean curvature flowSingularitiesWeak & singular flowsBoundaries & non-compact

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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. Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

    J. M. Daniels-Holgate and Or HershkovitsWider flows PDF

    Topics Boundaries & non-compact, Singularities · Same author J. M. Daniels-Holgate · Shared terms “conical singularities”, “asymptotically conical”, “isolated”

  2. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “asymptotically conical”

  3. Closed mean curvature flows with asymptotically conical singularities

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    Topics Weak & singular flows, Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “set flow”, “level set”

  4. Generic regularity of Level Set Flows with spherical singularity

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    Topics Weak & singular flows, Singularities · Shared terms “set flow”, “level set”

  5. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze

  6. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze

  7. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze

  8. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows PDF

    Topics Weak & singular flows, Singularities · Shared terms “set flow”, “level set”

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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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.