Paper · Oct 28, 2016

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

Wenlong Wang

math.DGarXiv:1610.09285v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove that all entire smooth strictly convex self-shrinking solutions on to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire self-shrinking solutions on to the Kähler-Ricci flow under certain conditions.

View PDF View HTML arXiv page
Submitted
Oct 28, 2016
Last revised
Jul 22, 2017 (v3)
Subjects
math.DG

Comments: Several typos revised

Ricci flowKähler–Ricci flowMean curvature flowLagrangian mean curvature 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 self-shrinking solutions to Kähler-Ricci flow on complex plane (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 Remarks on the self-shrinking Clifford torus (Christopher G. Evans, Jason D. Lotay and Felix Schulze, 2018)8Evans 2018 On the entire self-shrinking solutions to Lagrangian mean curvature flow II (Rongli Huang, Qianzhong Ou and Wenlong Wang, 2019)3Huang 2019 Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow (Arunima Bhattacharya and Jeremy Wall, 2024)7Bhattacharya 2024 Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II (Shanshan Li, Jiaru Lv and Rongli Huang, 2024)5Li 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. Rigidity of entire self-shrinking solutions to Kähler-Ricci flow with strictly real convex potential

    Wenlong Wang

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

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

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

  5. 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 “pseudo Euclidean”, “Euclidean space”

  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”, “entire smooth”, “rigidity theorem”

  7. 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 · Shared terms “Hessian”

  8. Remarks on the self-shrinking Clifford torus

    Christopher G. Evans, Jason D. Lotay and Felix SchulzeWider flows

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

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) 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)

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. 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.

  3. 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.