Paper · Nov 19, 2021

On the Stability of Cylindrical Singularities of the Mean Curvature Flow

Jingxuan Zhang

math.DGWider flowsarXiv:2111.10111
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Abstract

We study the rescaled mean curvature flow (MCF) of hypersurfaces that are global graphs over a fixed cylinder of arbitrary dimensions. We construct an explicit stable manifold for the rescaled MCF of finite codimensions in a suitable configuration space. For any initial hypersurface from this stable manifold, we construct a unique global solution to the rescaled MCF, and derive precise asymptotics for these solutions that are valid for all time. Using these asymptotics, we prove asymptotic stability of cylindrical singularities of arbitrary dimensions under generic initial perturbations. As a by-product, for any flow of hypersurfaces evolving according to the MCF that enters this stable manifold at any time and first develops a singularity at a subsequent time, we give a simple proof of the uniqueness of tangent flow, first established by Colding and Minicozzi. Moreover, in this case we show the unique singularity profile is determined by the hypersurface profile when the flow enters the stable manifold. For all results in this paper, there is no symmetry or solitonic assumption.

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Submitted
Nov 19, 2021
Subjects
math.DG, math.AP

Comments: 38 pp

Mean curvature flowSingularitiesStability & uniquenessLong-time existence & convergence

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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 mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)5Sun 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)8Lee 2023 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)6Chodosh 2023 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)2Ghosh 2025 Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow (Hanbing Fang and Yu Li, 2025)7Fang 2025 On the rate of convergence of cylindrical singularity in mean curvature flow (Yiqi Huang and Xinrui Zhao, 2025)1Huang 2025 Formation and structural stability of nondegenerate neckpinches (Kyeongsu Choi, Panagiota Daskalopoulos and Natasa Sesum, 2026)4Choi 2026 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)3Sun 2026
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  1. On the rate of convergence of cylindrical singularity in mean curvature flow

    Yiqi Huang and Xinrui ZhaoWider flows

    Topics Long-time existence & convergence, Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”, “rescaled mean”, “global graph”

  2. Cylindrical tangent flows in mean curvature flow

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    Topics Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”, “Minicozzi”, “tangent flow”

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

    Ao Sun and Jinxin XueWider flows

    Topics Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”, “perturbation”

  4. Formation and structural stability of nondegenerate neckpinches

    Kyeongsu Choi, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities, Stability & uniqueness · Shared terms “precise asymptotics”, “rescaled”, “Minicozzi”

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

    Ao Sun and Jinxin XueWider flows

    Topics Singularities · Shared terms “cylindrical singularities”, “rescaled mean”, “asymptotics”

  6. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

    Topics Singularities · Shared terms “generic initial”, “tangent flow”

  7. Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

    Hanbing Fang and Yu Li

    Topics Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”, “tangent flow”

  8. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows

    Topics Singularities, Stability & uniqueness · Shared terms “tangent flow”

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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