Paper · Jan 28, 2025

Passing through nondegenerate singularities in mean curvature flows

Ao Sun, Zhihan Wang, Jinxin Xue

math.DGWider flowsarXiv:2501.16678
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Abstract

In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new -distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the "decay order", a discrete mean curvature flow analog to Almgren's frequency function.

Comments: 52 pages, comments are welcome!

Mean curvature flowSingularitiesSurgery

Its neighbourhood

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 Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities (Ao Sun and Jinxin Xue, 2021)7Sun 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)6Sun 2022 Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)2Sun 2022 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)8Sun 2023 Regularity of cylindrical singular sets of mean curvature flow (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)3Sun 2025 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)5Székelyhidi 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)4Székelyhidi 2026 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)1Sun 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “nondegenerate singularities”, “cylindrical singularities”, “nondegenerate cylindrical”

  2. 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 “cylindrical singularities”, “nondegenerate cylindrical”, “isolated”

  3. Regularity of cylindrical singular sets of mean curvature flow

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Zhihan Wang, Jinxin Xue

  4. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities · Shared terms “nondegenerate cylindrical”, “cylindrical singularities”

  5. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities · Shared terms “nondegenerate”, “isolated”

  6. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue PDF

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

  7. Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

    Ao Sun and Jinxin Xue PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue

  8. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows PDF

    Topics Mean curvature flow · Same authors Ao Sun, Zhihan Wang

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).