Paper · Jul 11, 2021

Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

Ao Sun, Jinxin Xue

math.DGarXiv:2107.05066v2
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Abstract

This is the second paper in the series to study the generic dynamics of mean curvature flows. We study the initial perturbation of mean curvature flows, whose first singularity is modeled by an asymptotically conical shrinker. The noncompactness of the limiting shrinker creates essential difficulties. We introduce the Feynman-Kac formula to get precise asymptotic behaviour of the linearized rescaled mean curvature equation along an orbit. We also develop the invariant cone method for the noncompact setting for the local dynamics near the shrinker. As a consequence, we prove that after a generic initial perturbation, the perturbed rescaled mean curvature flow avoids the conical singularity.

Comments: 48 pages, 1 figure. Accepted by Sci. China Math

Mean curvature flowRicci solitonsSingularitiesLimit spaces & RCDBoundaries & 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 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)4Stolarski 2022 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)1Sun 2022 Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)3Sun 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)6Lee 2023 Regularity of cylindrical singular sets of mean curvature flow (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)8Sun 2025 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)2Chen 2026 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)5Sun 2026 On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis (Yu Li, Longhuan Yan and Wenjia Zhang, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue PDF

    Topics Singularities, Ricci solitons · Same authors Ao Sun, Jinxin Xue · Shared terms “generic dynamics”

  2. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al. PDF

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Same author Ao Sun · Shared terms “asymptotically conical”

  3. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

    Ao Sun and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue · Shared terms “rescaled mean”

  4. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

    Maxwell Stolarski PDF

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Shared terms “conical shrinker”, “asymptotically conical”, “precise asymptotic”

  5. Mean curvature flows with cylindrical singularities II: stability and genericity

    Ao Sun and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue · Shared terms “perturbation”

  6. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

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

  7. On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis

    Yu Li, Longhuan Yan and Wenjia Zhang PDF

    Topics Limit spaces & RCD, Boundaries & non-compact, Ricci solitons · Shared terms “asymptotically conical”, “perturbation”

  8. Regularity of cylindrical singular sets of mean curvature flow

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.