Paper · Oct 27, 2017

Diameter and curvature control under mean curvature flow

Panagiotis Gianniotis, Robert Haslhofer

math.DGWider flowsarXiv:1710.10347
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Abstract

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp -estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis of cylindrical regions (-tubes) under mean curvature flow. The results are new even in the most classical case of mean convex surfaces evolving by mean curvature flow in .

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Submitted
Oct 27, 2017
Subjects
math.DG
Mean curvature flowWeak & singular flows

Its neighbourhood

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  1. Bounded Diameter Under Mean Curvature Flow

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    Topics Weak & singular flows · Same author Robert Haslhofer · Shared terms “detailed”

  4. Lectures on mean curvature flow of surfaces

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    Topics Weak & singular flows · Same author Robert Haslhofer · Shared terms “detailed”, “intrinsic”

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

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  6. Quantitative avoidance for free boundary flows and applications

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    Topics Weak & singular flows · Same author Robert Haslhofer · Shared terms “mean convex”

  7. Ancient asymptotically cylindrical flows and applications

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  8. The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

    Yiqi Huang and Wenshuai JiangWider flows

    Topics Weak & singular flows · Shared terms “diameter”, “singular time”, “intrinsic”

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) 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)

In the guidebook

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

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

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.