Paper · Feb 28, 2017

Lagrangian Mean Curvature Flows and Moment maps

Hiroshi Konno

math.DGWider flowsarXiv:1703.00090v4
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Abstract

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples of Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.

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Submitted
Feb 28, 2017
Last revised
Nov 21, 2017 (v4)
Subjects
math.DG, math.SG

Comments: 34 pages, 1 figure; (ver. 2) The figure is fixed to be displayed correctly. Author's e-mail address is added.;(ver.3) A reference is added together with a related comment; (ver. 4) minor revision

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSelf-shrinkers & translatorsSingularities4-manifoldsKähler geometryHomogeneous spaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Local non-collapsing of volume for the Lagrangian mean curvature flow (Knut Smoczyk, 2018)1Smoczyk 2018 Complete Lagrangian self-shrinkers in mathbf R⁴ (Qing-Ming Cheng, Hiroaki Hori and Guoxin Wei, 2018)6Cheng 2018 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)2Evans 2019 On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow (Wei-Bo Su, 2020)7Su 2020 Cohomogeneity-One Lagrangian Mean Curvature Flow (Jesse Madnick and Albert Wood, 2022)3Madnick 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)4Lotay 2023 Generic neck pinch singularities along 2D Lagrangian mean curvature flow (Gábor Székelyhidi, 2026)8Székelyhidi 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)5Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local non-collapsing of volume for the Lagrangian mean curvature flow

    Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Self-shrinkers & translators · Shared terms “translating solitons”

  2. Lagrangian mean curvature flow with boundary

    Christopher G. Evans, Ben Lambert and Albert WoodWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Self-shrinkers & translators

  3. Cohomogeneity-One Lagrangian Mean Curvature Flow

    Jesse Madnick and Albert Wood

    Topics Lagrangian mean curvature flow, Homogeneous spaces, Self-shrinkers & translators · Shared terms “moment map”, “Lie groups”

  4. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, 4-manifolds

  5. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau

  6. Complete Lagrangian self-shrinkers in

    Qing-Ming Cheng, Hiroaki Hori and Guoxin WeiWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “Euclidean space”

  7. On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow

    Wei-Bo SuWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “translating solitons”

  8. Generic neck pinch singularities along 2D Lagrangian mean curvature flow

    Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  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.