Paper · Jun 7, 2024

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

Ping-Hung Lee, Chung-Jun Tsai

math.DGWider flowsarXiv:2406.04602
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Abstract

It is known that minimal Lagrangians in Kähler–Einstein manifolds of non-positive scalar curvature are linearly stable under Hamiltonian deformations. We prove that they are also stable under the Lagrangian mean curvature flow, and therefore establish the equivalence between linear stability and dynamical stability. Specifically, if one starts the mean curvature flow with a Lagrangian which is -close and Hamiltonian isotopic to a minimal Lagrangian, the flow exists smoothly for all time, and converges to that minimal Lagrangian. Due to the work of Neves [Ann. of Math. 2013], this cannot be true for -closeness.

Comments: 21 pages

Mean curvature flowLagrangian mean curvature flowEinstein metricsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability

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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 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)3Tsai 2023 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)8Su 2024 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)1Lee 2026 Almost isoclinic Lagrangian submanifolds (Tang-Kai Lee and Mu-Tao Wang, 2026)5Lee 2026 Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow (Ping-Hung Lee, 2026)6Lee 2026 Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds (Naotoshi Fujihara, Toru Kajigaya and Albert Wood, 2026)2Fujihara 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)7Flores 2026
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  1. Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Lagrangian mean curvature flow · Same authors Ping-Hung Lee, Chung-Jun Tsai · Shared terms “exists smoothly”, “flow exists”

  2. Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds

    Naotoshi Fujihara, Toru Kajigaya and Albert WoodWider flows PDF

    Topics Lagrangian mean curvature flow, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “minimal Lagrangian”

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

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

    Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai

  4. Mean Curvature Flows of Two-Convex Lagrangians

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

    Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai

  5. Almost isoclinic Lagrangian submanifolds

    Tang-Kai Lee and Mu-Tao WangWider flows PDF

    Topics Lagrangian mean curvature flow, Stability & uniqueness · Shared terms “minimal Lagrangian”

  6. Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

    Ping-Hung Lee PDF

    Topics Lagrangian mean curvature flow, Scalar curvature · Same author Ping-Hung Lee · Shared terms “minimal Lagrangian”, “flow exists”, “Specifically”

  7. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows PDF

    Topics Lagrangian mean curvature flow · Shared terms “minimal Lagrangian”

  8. Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

    Wei-Bo Su, Chung-Jun Tsai and Albert Wood PDF

    Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai

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) Lagrangian mean curvature flow Wider flows In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume… Start with: A canonical way to deform a Lagrangian submanifold (Smoczyk, 1996) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

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    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 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.