Paper · Feb 5, 2025

Quantitative uniqueness for mean curvature flow

Tobias Holck Colding, William P. Minicozzi

math.DGWider flowsarXiv:2502.03634
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Abstract

We show how to use the arguments of [CM2] to get a stronger effective version of uniqueness of blowups that has a number of consequences.

Mean curvature flowSingularitiesStability & uniqueness

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Connected papers

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 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Optimal growth bounds for eigenfunctions (Tobias Holck Colding and William P. Minicozzi, 2021)4Colding 2021 Singularities of Ricci flow and diffeomorphisms (Tobias Holck Colding and William P. Minicozzi, 2021)1Colding 2021 Singularities and diffeomorphisms (Tobias Holck Colding and William P. Minicozzi, 2022)3Colding 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)6Lee 2023 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)7Colding 2023 Propagation of symmetries for Ricci shrinkers (Tobias Holck Colding and William P. Minicozzi, 2023)8Colding 2023 A strong Frankel Theorem for shrinkers (Tobias Holck Colding and William P. Minicozzi, 2023)5Colding 2023 Distance between minimal surfaces and flows (Tobias Holck Colding and William P. Minicozzi, 2026)2Colding 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of Ricci flow and diffeomorphisms

    Tobias Holck Colding and William P. Minicozzi PDF

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

  2. Distance between minimal surfaces and flows

    Tobias Holck Colding and William P. MinicozziWider flows PDF

    Topics Mean curvature flow · Same authors Tobias Holck Colding, William P. Minicozzi

  3. Singularities and diffeomorphisms

    Tobias Holck Colding and William P. Minicozzi PDF

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

  4. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi PDF

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

  5. A strong Frankel Theorem for shrinkers

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Mean curvature flow · Same authors Tobias Holck Colding, William P. Minicozzi

  6. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Singularities, Stability & uniqueness

  7. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Same authors Tobias Holck Colding, William P. Minicozzi

  8. Propagation of symmetries for Ricci shrinkers

    Tobias Holck Colding and William P. Minicozzi PDF

    Same authors Tobias Holck Colding, William P. Minicozzi

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. 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) 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.