Paper · Feb 12, 2020

Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

Ryosuke Takahashi

math.DGWider flowsarXiv:2002.05132v6
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Abstract

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated -forms that fits into the GIT framework for dHYM metrics recently discovered by Collins-Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a -subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.

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Submitted
Feb 12, 2020
Last revised
Oct 15, 2020 (v6)
Subjects
math.DG

Comments: 24 pages, final version, to appear in Internat. J. Math

Mean curvature flowLagrangian mean curvature flowLong-time existence & convergence

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

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  3. Convergence of mean curvature flow in hyperkähler manifolds

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  5. Collapsing of the line bundle mean curvature flow on Kähler surfaces

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    Topics Long-time existence & convergence · Same author Ryosuke Takahashi · Shared terms “bundle mean”, “line bundle”, “deformed Hermitian”

  6. Mean Curvature Flows of Two-Convex Lagrangians

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

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    Topics Lagrangian mean curvature flow, Long-time existence & convergence

  8. Ancient solutions and translators of Lagrangian mean curvature flow

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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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.