Paper · Apr 3, 2017

Non-fattening of mean curvature flow at singularities of mean convex type

Or Hershkovits, Brian White

math.DGWider flowsarXiv:1704.00431v3
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Abstract

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. This generalizes the well known fact that the level set flow of a mean convex initial hypersurface M does not fatten. This also provides the first instance where non-fattening is concluded from local information around the singular set.

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Submitted
Apr 3, 2017
Last revised
Feb 4, 2018 (v3)
Subjects
math.DG
Journal
Comm. Pure Appl. Math. 73 (2020), no. 3, 558-580

Comments: 19 pages. Some new material (see definition 10 and theorem 11) clarifying the nature of the hypotheses has been added

Mean curvature flowSingularitiesWeak & singular flows

Its neighbourhood

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    Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al.Wider flows

    Topics Weak & singular flows, Singularities · Same authors Or Hershkovits, Brian White · Shared terms “mean convex”, “Specifically”

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    Topics Singularities · Same author Brian White · Shared terms “fattening”, “smoothly embedded”, “singular point”

  3. Mean Curvature Flow from Conical Singularities

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  4. Ancient low entropy flows, mean convex neighborhoods, and uniqueness

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    Topics Weak & singular flows, Singularities · Same author Or Hershkovits · Shared terms “mean convex”, “information”

  5. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

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

  6. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows

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

  7. Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

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  8. Regularity and stability results for the level set flow via the mean curvature flow with surgery

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

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) 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) 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)

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

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  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.