Paper · Feb 2, 2019

Mean curvature flow of surfaces in a hyperkähler -manifold

Hongbing Qiu, Linlin Sun

math.DGWider flowsarXiv:1902.00645v2
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we firstly prove that every hyper-Lagrangian submanifold in a hyperkähler -manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the complex phase map of self-shrinking surfaces in . Last but not least, by using the previous rigidity result, we show that the mean curvature flow from a closed surface with the image of the complex phase map contained in in a hyperkähler -manifold does not develop any Type \Rmn1 singularity.

View PDF View HTML arXiv page
Submitted
Feb 2, 2019
Last revised
Nov 24, 2020 (v2)
Subjects
math.DG
Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSelf-shrinkers & translatorsSingularitiesKähler geometryStability & uniqueness

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)7Smoczyk 2016 Lagrangian Mean Curvature Flows and Moment maps (Hiroshi Konno, 2017)5Konno 2017 Local non-collapsing of volume for the Lagrangian mean curvature flow (Knut Smoczyk, 2018)4Smoczyk 2018 Convergence of mean curvature flow in hyperkähler manifolds (Keita Kunikawa and Ryosuke Takahashi, 2018)1Kunikawa 2018 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)6Lotay 2023 Rigidity theorems for complete Legendrian self-shrinkers (Shu-Cheng Chang, Hongbing Qiu and Liuyang Zhang, 2025)2Chang 2025 Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2026)8Bhattacharya 2026 Almost isoclinic Lagrangian submanifolds (Tang-Kai Lee and Mu-Tao Wang, 2026)3Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of mean curvature flow in hyperkähler manifolds

    Keita Kunikawa and Ryosuke TakahashiWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “Lagrangian submanifold”, “hyperkähler”, “complex Lagrangian”

  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, Stability & uniqueness · Same author Hongbing Qiu · Shared terms “rigidity theorem”, “self shrinking”

  3. Almost isoclinic Lagrangian submanifolds

    Tang-Kai Lee and Mu-Tao WangWider flows

    Topics Lagrangian mean curvature flow, Stability & uniqueness · Shared terms “Lagrangian submanifold”

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

    Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Self-shrinkers & translators

  5. Lagrangian Mean Curvature Flows and Moment maps

    Hiroshi KonnoWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Self-shrinkers & translators

  6. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “hyperkähler”

  7. 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 · Shared terms “Lagrangian submanifold”, “demonstrate”

  8. Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “Lagrangian submanifold”

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.