Paper · Jan 22, 2018

Local non-collapsing of volume for the Lagrangian mean curvature flow

Knut Smoczyk

math.DGWider flowsarXiv:1801.07303
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As on arXiv

Abstract

We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi-Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in that evolve by this reparametrized flow

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Submitted
Jan 22, 2018
Subjects
math.DG
Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSelf-shrinkers & translators(Non)collapsingKähler geometry

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 Generalized Lagrangian mean curvature flows: the cotangent bundle case (Knut Smoczyk, Mao-Pei Tsui and Mu-Tao Wang, 2016)2Smoczyk 2016 Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature (Keita Kunikawa, 2016)7Kunikawa 2016 Lagrangian Mean Curvature Flows and Moment maps (Hiroshi Konno, 2017)4Konno 2017 Lagrangian mean curvature flow of Whitney spheres (Andreas Savas-Halilaj and Knut Smoczyk, 2018)1Savas-Halila 2018 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)8Evans 2019 On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow (Wei-Bo Su, 2020)3Su 2020 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)5Lotay 2022 Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class (Xiang Li, Yong Luo and Jun Sun, 2024)6Li 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Lagrangian mean curvature flow of Whitney spheres

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Knut Smoczyk

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

    Knut Smoczyk, Mao-Pei Tsui and Mu-Tao WangWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Same author Knut Smoczyk

  3. 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 “Lagrangian translating”, “translating solitons”

  4. Lagrangian Mean Curvature Flows and Moment maps

    Hiroshi KonnoWider flows

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

  5. Ancient solutions and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

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

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

    Xiang Li, Yong Luo and Jun SunWider flows

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

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

    Keita KunikawaWider flows

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

  8. 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

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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.