Paper · Apr 26, 2016

Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow

Jingyi Chen, John Man Shun Ma

math.DGWider flowsarXiv:1604.07563
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Abstract

We study geometric properties of the Lagrangian self-shrinking tori in . When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in. When the area bound is small, we show that any Lagrangian self-shrinking torus in with small area is embedded with uniform curvature estimates, and the space of such tori is compact. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in , along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in in.

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Submitted
Apr 26, 2016
Subjects
math.DG

Comments: 33 pages

Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesLimit spaces & RCD

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 Remarks on the self-shrinking Clifford torus (Christopher G. Evans, Jason D. Lotay and Felix Schulze, 2018)8Evans 2018 Complete Lagrangian self-shrinkers in mathbf R⁴ (Qing-Ming Cheng, Hiroaki Hori and Guoxin Wei, 2018)2Cheng 2018 Lagrangian mean curvature flow of Whitney spheres (Andreas Savas-Halilaj and Knut Smoczyk, 2018)3Savas-Halila 2018 On the entire self-shrinking solutions to Lagrangian mean curvature flow II (Rongli Huang, Qianzhong Ou and Wenlong Wang, 2019)1Huang 2019 Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori (Yakov Berchenko-Kogan, 2020)5Berchenko-Ko 2020 On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow (Wei-Bo Su, 2020)7Su 2020 Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class (Xiaoli Han, Jiayu Li and Jun Sun, 2024)6Han 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)4Ghosh 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the entire self-shrinking solutions to Lagrangian mean curvature flow II

    Rongli Huang, Qianzhong Ou and Wenlong WangWider flows

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

  2. Complete Lagrangian self-shrinkers in

    Qing-Ming Cheng, Hiroaki Hori and Guoxin WeiWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators

  3. Lagrangian mean curvature flow of Whitney spheres

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Singularities

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

    Sourav GhoshWider flows

    Topics Lagrangian mean curvature flow, Singularities

  5. Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori

    Yakov Berchenko-KoganWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “shrinking tori”, “self shrinking”, “upper bound”

  6. Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class

    Xiaoli Han, Jiayu Li and Jun SunWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Singularities

  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

  8. Remarks on the self-shrinking Clifford torus

    Christopher G. Evans, Jason D. Lotay and Felix SchulzeWider flows

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

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) 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) 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)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.