Paper · Feb 7, 2018

Complete Lagrangian self-shrinkers in

Qing-Ming Cheng, Hiroaki Hori, Guoxin Wei

math.DGWider flowsarXiv:1802.02396v2
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Abstract

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space with constant squared norm of the second fundamental form.

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Submitted
Feb 7, 2018
Last revised
May 9, 2018 (v2)
Subjects
math.DG

Comments: 53 pages

Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSurfaces

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 Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow (Jingyi Chen and John Man Shun Ma, 2016)3Chen 2016 Lagrangian Mean Curvature Flows and Moment maps (Hiroshi Konno, 2017)6Konno 2017 Stability properties of complete self-shrinking surfaces in ℝ³ (Hilário Alencar, Gregório Silva Neto and Detang Zhou, 2021)5Alencar 2021 Embedded cylindrical and doughnut-shaped λ-hypersurfaces (Qing-Ming Cheng, Junqi Lai and Guoxin Wei, 2024)1Cheng 2024 Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II (Shanshan Li, Jiaru Lv and Rongli Huang, 2024)7Li 2024 Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class (Xiang Li, Yong Luo and Jun Sun, 2024)8Li 2024 Rigidity theorems for complete Legendrian self-shrinkers (Shu-Cheng Chang, Hongbing Qiu and Liuyang Zhang, 2025)2Chang 2025 The rigidity theorems for complete self-expander of mean curvature flow (Zhi Li and Guoxin Wei, 2026)4Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Embedded cylindrical and doughnut-shaped -hypersurfaces

    Qing-Ming Cheng, Junqi Lai and Guoxin WeiWider flows

    Topics Self-shrinkers & translators · Same authors Qing-Ming Cheng, Guoxin Wei

  2. Rigidity theorems for complete Legendrian self-shrinkers

    Shu-Cheng Chang, Hongbing Qiu and Liuyang ZhangWider flows

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

  3. 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

  4. The rigidity theorems for complete self-expander of mean curvature flow

    Zhi Li and Guoxin WeiWider flows

    Topics Self-shrinkers & translators · Same author Guoxin Wei · Shared terms “complete self”, “fundamental form”, “Euclidean space”

  5. Stability properties of complete self-shrinking surfaces in

    Hilário Alencar, Gregório Silva Neto and Detang ZhouWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “complete self”, “Euclidean space”

  6. Lagrangian Mean Curvature Flows and Moment maps

    Hiroshi KonnoWider flows

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

  7. 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 · Shared terms “Euclidean space”

  8. 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, Surfaces, 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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  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.