Paper · Sep 18, 2017

A pointwise approach to rigidity of almost graphical self-shrinking solutions of mean curvature flows

Dongsheng Li, Yingfeng Xu, Yu Yuan

math.AParXiv:1709.05902
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As on arXiv

Abstract

We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.

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Submitted
Sep 18, 2017
Subjects
math.AP, math.DG

Comments: 5 pages, Analysis of PDEs, Differential Geometry

Mean curvature flowLagrangian mean curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniquenessBoundaries & non-compact

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 Rigidity of entire self-shrinking solutions to Kähler-Ricci flow with strictly real convex potential (Wenlong Wang, 2016)2Wang 2016 Rigidity of entire self-shrinking solutions to Kähler-Ricci flow on complex plane (Wenlong Wang, 2016)4Wang 2016 Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows (Wenlong Wang, 2016)3Wang 2016 On the entire self-shrinking solutions to Lagrangian mean curvature flow II (Rongli Huang, Qianzhong Ou and Wenlong Wang, 2019)1Huang 2019 Optimal regularity for Lagrangian mean curvature type equations (Arunima Bhattacharya and Ravi Shankar, 2020)7Bhattacharya 2020 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)8Lotay 2022 Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow (Arunima Bhattacharya and Jeremy Wall, 2024)5Bhattacharya 2024 Rigidity theorems for complete Legendrian self-shrinkers (Shu-Cheng Chang, Hongbing Qiu and Liuyang Zhang, 2025)6Chang 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 “shrinking solutions”, “self shrinking”

  2. Rigidity of entire self-shrinking solutions to Kähler-Ricci flow with strictly real convex potential

    Wenlong Wang

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

  3. Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows

    Wenlong Wang

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

  4. Rigidity of entire self-shrinking solutions to Kähler-Ricci flow on complex plane

    Wenlong Wang

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “shrinking solutions”, “self shrinking”

  5. Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

    Arunima Bhattacharya and Jeremy Wall

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Ricci solitons

  6. Rigidity theorems for complete Legendrian self-shrinkers

    Shu-Cheng Chang, Hongbing Qiu and Liuyang ZhangWider flows

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

  7. Optimal regularity for Lagrangian mean curvature type equations

    Arunima Bhattacharya and Ravi Shankar

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

  8. 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 “Lagrangian angle”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 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.