Paper · Aug 21, 2018

Convergence of mean curvature flow in hyperkähler manifolds

Keita Kunikawa, Ryosuke Takahashi

math.DGWider flowsarXiv:1808.06997v4
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Abstract

Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called the "twistor energy" by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex Lagrangian submanifold for one of the hyperkähler complex structure. In particular, our result implies some kind of energy gap theorem for hyperkähler manifolds which have no complex Lagrangian submanifolds.

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Submitted
Aug 21, 2018
Last revised
Nov 21, 2019 (v4)
Subjects
math.DG
Journal
Pacific J. Math. 305 (2020) 667-691
DOI
10.2140/pjm.2020.305.667

Comments: 21 pages, 1 figure. Minor corrections and change of title. Final version, to appear in Pacific J. Math

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauKähler geometryLong-time existence & convergence

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  5. Lagrangian mean curvature flow in the Kummer K3 surface

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    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence

  6. Mean Curvature Flows of Two-Convex Lagrangians

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    Topics Lagrangian mean curvature flow, Long-time existence & convergence · Shared terms “Lagrangian submanifold”

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

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

  8. Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

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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.