Paper · Mar 6, 2019

Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature

Xiaolong Li, Lei Ni

math.DGarXiv:1903.02615
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Abstract

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new feature is that no curvature upper bound is assumed.

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Submitted
Mar 6, 2019
Subjects
math.DG

Comments: 18 pages. Comments are welcome

Ricci flowKähler–Ricci flowRicci solitonsAncient solutionsSphere theoremsKähler geometry

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 Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature (Xiaolong Li, Lei Ni and Kui Wang, 2016)3Li 2016 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)6Li 2018 Kähler manifolds and cross quadratic bisectional curvature (Lei Ni and Fangyang Zheng, 2019)4Ni 2019 Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature (Shijin Zhang, 2019)1Zhang 2019 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)5Lee 2019 Ancient solutions to the Ricci flow with isotropic curvature conditions (Jae Ho Cho and Yu Li, 2020)7Cho 2020 Ancient solutions to the Kähler Ricci flow (Yu Li, 2020)2Li 2020 Kähler-Ricci Flow preserves negative anti-bisectional curvature (Gabriel Khan and Fangyang Zheng, 2020)8Khan 2020
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature

    Shijin Zhang

    Topics Kähler geometry, Ricci solitons · Shared terms “orthogonal bisectional”, “nonnegative orthogonal”, “bisectional curvature”

  2. Ancient solutions to the Kähler Ricci flow

    Yu Li

    Topics Kähler–Ricci flow, Ancient solutions, Kähler geometry · Shared terms “bisectional curvature”

  3. Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature

    Xiaolong Li, Lei Ni and Kui Wang

    Topics Sphere theorems, Ricci solitons · Same authors Xiaolong Li, Lei Ni · Shared terms “gradient shrinking”, “shrinking Ricci”

  4. Kähler manifolds and cross quadratic bisectional curvature

    Lei Ni and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Same author Lei Ni · Shared terms “bisectional curvature”

  5. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “orthogonal bisectional”, “nonnegative orthogonal”, “bisectional curvature”

  6. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Sphere theorems, Ancient solutions, Ricci solitons · Same author Xiaolong Li

  7. Ancient solutions to the Ricci flow with isotropic curvature conditions

    Jae Ho Cho and Yu Li

    Topics Sphere theorems, Kähler–Ricci flow, Ancient solutions · Shared terms “weakly PIC”

  8. Kähler-Ricci Flow preserves negative anti-bisectional curvature

    Gabriel Khan and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “bisectional curvature”, “orthogonal”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

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