Paper · Feb 1, 2016

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

Wenlong Wang

math.DGarXiv:1602.00368v2
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Abstract

We show that every entire self-shrinking solution on to the Kähler-Ricci flow must be generated from a quadratic potential.

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Submitted
Feb 1, 2016
Last revised
Oct 24, 2016 (v2)
Subjects
math.DG
Ricci flowKähler–Ricci flowSelf-shrinkers & translatorsKähler 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 Rigidity of entire self-shrinking solutions to Kähler-Ricci flow with strictly real convex potential (Wenlong Wang, 2016)1Wang 2016 Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows (Wenlong Wang, 2016)2Wang 2016 A pointwise approach to rigidity of almost graphical self-shrinking solutions of mean curvature flows (Dongsheng Li, Yingfeng Xu and Yu Yuan, 2017)4Li 2017 A note on compact κ-solutions of Kähler-Ricci flow (Yuxing Deng and Xiaohua Zhu, 2018)8Deng 2018 On the entire self-shrinking solutions to Lagrangian mean curvature flow II (Rongli Huang, Qianzhong Ou and Wenlong Wang, 2019)3Huang 2019 Compactness and rigidity of self-shrinking surfaces (Tang-Kai Lee, 2021)6Lee 2021 Entire self-expanders for power of σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)5Wang 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)7Wu 2022
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 Kähler–Ricci flow, Self-shrinkers & translators, Kähler geometry · Same author Wenlong Wang · Shared terms “entire self”, “shrinking solutions”, “self shrinking”

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

    Wenlong Wang

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

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

    Rongli Huang, Qianzhong Ou and Wenlong WangWider flows

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

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

    Dongsheng Li, Yingfeng Xu and Yu Yuan

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

  5. Entire self-expanders for power of curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators · Shared terms “entire self”

  6. Compactness and rigidity of self-shrinking surfaces

    Tang-Kai LeeWider flows

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

  7. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang

    Topics Kähler–Ricci flow, Kähler geometry

  8. A note on compact -solutions of Kähler-Ricci flow

    Yuxing Deng and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.