Paper · Oct 16, 2017

Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

Toru Kajigaya, Keita Kunikawa

math.DGWider flowsarXiv:1710.05537v3
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Abstract

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler-Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form by using the methodology proposed by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian submanifold in a Kähler manifold and investigate the variational problem of for the weighted volume functional. We call a stationary point of the weighted volume functional -minimal, and define the notion of Hamiltonian -stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk-Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an -minimal and Hamiltonian -stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an -minimal Lagrangian submanifold.

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Submitted
Oct 16, 2017
Last revised
Mar 19, 2018 (v3)
Subjects
math.DG
DOI
10.1016/j.geomphys.2018.02.011

Comments: 40 pages; (ver.3) minor corrections. (ver.2) Tex file format is changed, Corollary 2.5 and a reference are added, minor corrections

Mean curvature flowLagrangian mean curvature flowEinstein metricsKähler geometryStability & uniquenessLong-time existence & convergenceKähler–Einstein & K-stability

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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) 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 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.