Paper · Jul 22, 2025

Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

J. M. Daniels-Holgate, Or Hershkovits

math.DGWider flowsarXiv:2507.16805v2
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Abstract

In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces and encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time , and if then for every . This is seemingly the first backwards uniqueness result for any geometric flow with singularities, that assumes neither self-shrinking nor global asymptotically conical behaviour. This necessitates the development of new global tools to deal with both the core of the singularity, its asymptotic structure, and the smooth part of the flows simultaneously. As an immediate application, we show that low entropy flows in are backwards unique

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & non-compact

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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 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)1Lee 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)2Chodosh 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)6Lee 2024 Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds (Dasong Li and John Man Shun Ma, 2024)8Li 2024 How close is too close for singular mean curvature flows? (Joshua Daniels-Holgate and Or Hershkovits, 2025)4Daniels-Holg 2025 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)5Xie 2025 Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2025)7Gama 2025 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)3Daniels-Holg 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

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

  2. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows PDF

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

  3. Generic mean curvature flow has finitely many singularities

    J. M. Daniels-Holgate, Or Hershkovits and Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Same authors J. M. Daniels-Holgate, Or Hershkovits · Shared terms “compact hypersurfaces”, “encounter”, “multiplicity”

  4. How close is too close for singular mean curvature flows?

    Joshua Daniels-Holgate and Or HershkovitsWider flows PDF

    Topics Singularities · Same authors J. M. Daniels-Holgate, Or Hershkovits · Shared terms “multiplicity”, “demonstrate”

  5. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “asymptotically conical”

  6. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao PDF

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

  7. Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact, Singularities

  8. Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds

    Dasong Li and John Man Shun MaWider flows PDF

    Topics Boundaries & non-compact, Stability & uniqueness · Shared terms “backwards uniqueness”, “geometric flows”, “compact hypersurfaces”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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.