Paper · Apr 16, 2019

On the entire self-shrinking solutions to Lagrangian mean curvature flow II

Rongli Huang, Qianzhong Ou, Wenlong Wang

math.DGWider flowsarXiv:1904.07713
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Abstract

We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in . The proofs rely on a priori estimates and barriers construction.

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Submitted
Apr 16, 2019
Subjects
math.DG, math.AP

Comments: 16 pages

Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translators

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)1Wang 2016 Rigidity of entire self-shrinking solutions to Kähler-Ricci flow on complex plane (Wenlong Wang, 2016)2Wang 2016 Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow (Jingyi Chen and John Man Shun Ma, 2016)6Chen 2016 Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows (Wenlong Wang, 2016)3Wang 2016 A pointwise approach to rigidity of almost graphical self-shrinking solutions of mean curvature flows (Dongsheng Li, Yingfeng Xu and Yu Yuan, 2017)5Li 2017 On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow (Wei-Bo Su, 2020)7Su 2020 A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase (Arunima Bhattacharya and Jeremy Wall, 2024)8Bhattacharya 2024 Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II (Shanshan Li, Jiaru Lv and Rongli Huang, 2024)4Li 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same author Wenlong Wang · Shared terms “entire self”, “shrinking solutions”, “self shrinking”

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

    Wenlong Wang

    Topics Self-shrinkers & translators · Same author Wenlong Wang · Shared terms “entire self”, “shrinking solutions”, “self shrinking”

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

    Wenlong Wang

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Wenlong Wang · Shared terms “shrinking solutions”, “entire self”, “self shrinking”

  4. Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

    Shanshan Li, Jiaru Lv and Rongli HuangWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Rongli Huang

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

    Dongsheng Li, Yingfeng Xu and Yu Yuan

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

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

    Jingyi Chen and John Man Shun MaWider flows

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

  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. A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

    Arunima Bhattacharya and Jeremy WallWider flows

    Topics Lagrangian mean curvature flow · Shared terms “priori estimates”

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)

In the guidebook

  1. 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.