Paper · Oct 1, 2017

A strong stability condition on minimal submanifolds and its implications

Chung-Jun Tsai, Mu-Tao Wang

math.DGWider flowsarXiv:1710.00433v2
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Abstract

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a C^1 neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [arXiv:1605.03645] which applies only to calibrated submanifolds of special holonomy ambient manifolds.

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Submitted
Oct 1, 2017
Last revised
Dec 6, 2018 (v2)
Subjects
math.DG

Comments: 50 pages; Appendix C to gives the details for the C^2 convergence; to appear in J. Reine Angew. Math. (Crelle's Journal)

Mean curvature flowRicci-flat & Calabi–YauStability & uniquenessLong-time existence & convergenceMinimal & CMC surfaces

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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 2016, 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 Generalized Lagrangian mean curvature flows: the cotangent bundle case (Knut Smoczyk, Mao-Pei Tsui and Mu-Tao Wang, 2016)7Smoczyk 2016 Mean curvature flows in manifolds of special holonomy (Chung-Jun Tsai and Mu-Tao Wang, 2016)1Tsai 2016 Consequences of strong stability of minimal submanifolds (Jason D. Lotay and Felix Schulze, 2018)2Lotay 2018 A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)6Tsai 2023 Mean Curvature Flows of Two-Convex Lagrangians (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)5Tsai 2023 Entire solutions of two-convex Lagrangian mean curvature flows (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)4Tsai 2023 Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature (Ping-Hung Lee and Chung-Jun Tsai, 2024)8Lee 2024 Stability of Hyperkähler Flow (Kuan-Hui Lee, 2026)3Lee 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 in manifolds of special holonomy

    Chung-Jun Tsai and Mu-Tao WangWider flows

    Topics Minimal & CMC surfaces, Ricci-flat & Calabi–Yau, Stability & uniqueness · Same authors Chung-Jun Tsai, Mu-Tao Wang · Shared terms “minimal submanifolds”, “special holonomy”, “calibrated submanifolds”

  2. Consequences of strong stability of minimal submanifolds

    Jason D. Lotay and Felix SchulzeWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness · Shared terms “Strong Stability”, “minimal submanifolds”, “dynamical stability”

  3. Stability of Hyperkähler Flow

    Kuan-Hui LeeWider flows

    Topics Minimal & CMC surfaces, Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “stability theorem”, “uniqueness theorem”

  4. Entire solutions of two-convex Lagrangian mean curvature flows

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows

    Topics Long-time existence & convergence · Same authors Chung-Jun Tsai, Mu-Tao Wang

  5. Mean Curvature Flows of Two-Convex Lagrangians

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows

    Topics Long-time existence & convergence · Same authors Chung-Jun Tsai, Mu-Tao Wang

  6. A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows

    Topics Mean curvature flow · Same authors Chung-Jun Tsai, Mu-Tao Wang

  7. Generalized Lagrangian mean curvature flows: the cotangent bundle case

    Knut Smoczyk, Mao-Pei Tsui and Mu-Tao WangWider flows

    Topics Ricci-flat & Calabi–Yau, Long-time existence & convergence, Stability & uniqueness · Same author Mu-Tao Wang

  8. Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature

    Ping-Hung Lee and Chung-Jun TsaiWider flows

    Topics Stability & uniqueness · Same author Chung-Jun Tsai · Shared terms “dynamical stability”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.