Paper · Apr 11, 2016

Generalized Lagrangian mean curvature flows: the cotangent bundle case

Knut Smoczyk, Mao-Pei Tsui, Mu-Tao Wang

math.DGWider flowsarXiv:1604.02936
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Abstract

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0) form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

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Submitted
Apr 11, 2016
Subjects
math.DG

Comments: 31 pages

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauKähler geometryStability & uniquenessLong-time existence & convergenceHarmonic maps & DeTurck trick

Its neighbourhood

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  2. Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

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    Topics Lagrangian mean curvature flow, Stability & uniqueness · Shared terms “cotangent bundle”, “generalized Lagrangian”, “Lagrangian submanifold”

  3. Mean Curvature Flows of Two-Convex Lagrangians

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  4. Almost isoclinic Lagrangian submanifolds

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  5. Local non-collapsing of volume for the Lagrangian mean curvature flow

    Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Same author Knut Smoczyk

  6. Mean curvature flows in manifolds of special holonomy

    Chung-Jun Tsai 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”

  7. Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

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    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence · Shared terms “generalized Lagrangian”, “generalized mean”, “Lagrangian submanifold”

  8. Lagrangian mean curvature flow of Whitney spheres

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    Topics Lagrangian mean curvature flow · Same author Knut Smoczyk

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) 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) 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.