Paper · Jun 8, 2016

Lagrangian Flows, Maslov Index Zero and Special Lagrangians

Andrew A. Cooper, Jon Wolfson

math.DGWider flowsarXiv:1606.02691v3
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Abstract

We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian cycles (possibly with differing phases). We use the flow of cycles to obtain the fact that special lagrangian cycles generate the part of the lagrangian homology which lies in the image of the Hurewicz homomorphism. We also establish a weak version of a conjecture of Thomas-Yau regarding lagrangian mean curvature flow.

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Submitted
Jun 8, 2016
Last revised
Sep 15, 2016 (v3)
Subjects
math.DG

Comments: v2: Extended Corollary 0.5 from to , namely: The lagrangian homology of any simply-connected closed Calabi-Yau manifold is generated by special lagrangian cycles (possibly with different phases), in dimensions v3: Clarified the class of varifolds we use. Added details to some arguments in sections 3 and 5

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSingularitiesKähler geometryWeak & singular flows

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    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Weak & singular flows · Shared terms “Maslov”, “Thomas Yau”

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    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “special Lagrangian”, “Thomas Yau”

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    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Maslov”

  5. Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class

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    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Maslov”

  6. Lagrangian mean curvature flow of surfaces with mean curvature bound

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    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Maslov”

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

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    Topics Lagrangian mean curvature flow, Singularities · Shared terms “special Lagrangian”, “infinite time”, “Thomas”

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    Topics Lagrangian mean curvature flow, Weak & singular flows, Singularities

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.