Paper · May 12, 2016

Mean curvature flows in manifolds of special holonomy

Chung-Jun Tsai, Mu-Tao Wang

math.DGWider flowsarXiv:1605.03645v2
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Abstract

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant–Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

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Submitted
May 12, 2016
Last revised
Oct 12, 2016 (v2)
Subjects
math.DG

Comments: typos corrected; small changes reflecting referee comments; remark 1.5 added; to appear in J. Differential Geom

Mean curvature flowEinstein metricsRicci-flat & Calabi–YauKähler geometryStability & uniquenessMinimal & 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)2Smoczyk 2016 A strong stability condition on minimal submanifolds and its implications (Chung-Jun Tsai and Mu-Tao Wang, 2017)1Tsai 2017 Mirror of volume functionals on manifolds with special holonomy (Kotaro Kawai and Hikaru Yamamoto, 2021)7Kawai 2021 A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)3Tsai 2023 Mean Curvature Flows of Two-Convex Lagrangians (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)4Tsai 2023 Entire solutions of two-convex Lagrangian mean curvature flows (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)5Tsai 2023 Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature (Ping-Hung Lee and Chung-Jun Tsai, 2024)6Lee 2024 Stability of Hyperkähler Flow (Kuan-Hui Lee, 2026)8Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A strong stability condition on minimal submanifolds and its implications

    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. Generalized Lagrangian mean curvature flows: the cotangent bundle case

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

    Topics Ricci-flat & Calabi–Yau, Kähler geometry, Stability & uniqueness · Same author Mu-Tao Wang · Shared terms “cotangent bundle”, “zero section”

  3. 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 · Shared terms “complex projective”, “projective space”

  4. Mean Curvature Flows of Two-Convex Lagrangians

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

    Topics Mean curvature flow · Same authors Chung-Jun Tsai, Mu-Tao Wang · Shared terms “proof relies”

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

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

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

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

    Ping-Hung Lee and Chung-Jun TsaiWider flows

    Topics Einstein metrics, Kähler geometry, Stability & uniqueness · Same author Chung-Jun Tsai

  7. Mirror of volume functionals on manifolds with special holonomy

    Kotaro Kawai and Hikaru YamamotoWider flows

    Topics Ricci-flat & Calabi–Yau, Kähler geometry, Stability & uniqueness · Shared terms “special holonomy”

  8. Stability of Hyperkähler Flow

    Kuan-Hui LeeWider flows

    Topics Minimal & CMC surfaces, Ricci-flat & Calabi–Yau, Kähler geometry

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.