Paper · Apr 30, 2019

Regularity of elliptic and parabolic systems

Tobias Holck Colding, William P. Minicozzi

math.DGWider flowsarXiv:1905.00085v3
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Abstract

We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.

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Submitted
Apr 30, 2019
Last revised
Feb 14, 2020 (v3)
Subjects
math.DG, math.AP
Mean curvature flowSingularitiesStability & uniqueness

Its neighbourhood

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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. Complexity of parabolic systems

    Tobias Holck Colding and William P. Minicozzi

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  3. Entropy and codimension bounds for generic singularities

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  4. Liouville properties

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  5. Singularities of Ricci flow and diffeomorphisms

    Tobias Holck Colding and William P. Minicozzi

    Topics Singularities, Stability & uniqueness · Same authors Tobias Holck Colding, William P. Minicozzi

  6. Distance between minimal surfaces and flows

    Tobias Holck Colding and William P. MinicozziWider flows

    Topics Mean curvature flow · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “elliptic”

  7. Singularities and diffeomorphisms

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  8. Optimal growth bounds for eigenfunctions

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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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.