Paper · Apr 19, 2024

Expanding Ricci solitons coming out of weakly PIC1 metric cones

Pak-Yeung Chan, Man-Chun Lee, Luke T. Peachey

math.DGarXiv:2404.12755v2
PDF Abstract page

As on arXiv

Abstract

Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 Kähler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.

Comments: 21 pages, minor changes, all comments are welcome

Ricci flowRicci solitonsSphere theoremsKähler geometryWeak & singular flows

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 2021, 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 Volume growth estimates of gradient Ricci solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2022)4Chan 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)2Lee 2022 Gap Theorem on Riemannian manifolds using Ricci flow (Pak-Yeung Chan and Man-Chun Lee, 2023)6Chan 2023 Ricci flow and PIC1 (Peter M. Topping, 2023)7Topping 2023 Rigidity of area non-increasing maps (Man-Chun Lee, Luen-Fai Tam and Jingbo Wan, 2023)8Lee 2023 On the weakly conical expanding gradient Ricci solitons (Pak-Yeung Chan and Man-Chun Lee, 2025)1Chan 2025 Steady Gradient Ricci Solitons with O(p)× O(q) Symmetry (Lucas Lavoyer and Luke T. Peachey, 2025)3Lavoyer 2025 Higher-dimensional flying wing Steady Ricci Solitons (Pak-Yeung Chan, Yi Lai and Man-Chun Lee, 2025)5Chan 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the weakly conical expanding gradient Ricci solitons

    Pak-Yeung Chan and Man-Chun Lee PDF

    Topics Ricci solitons · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “expanding gradient”, “gradient Ricci”, “Deruelle”

  2. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Sphere theorems · Same author Man-Chun Lee · Shared terms “PIC”, “Deruelle Schulze”, “Schulze Simon”

  3. Steady Gradient Ricci Solitons with Symmetry

    Lucas Lavoyer and Luke T. Peachey PDF

    Topics Sphere theorems, Ricci solitons · Same author Luke T. Peachey · Shared terms “gradient Ricci”

  4. Volume growth estimates of gradient Ricci solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ricci solitons · Same author Pak-Yeung Chan · Shared terms “volume growth”, “expanding gradient”, “gradient Ricci”

  5. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee PDF

    Topics Ricci solitons · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “Deruelle”, “gradient Ricci”

  6. Gap Theorem on Riemannian manifolds using Ricci flow

    Pak-Yeung Chan and Man-Chun Lee PDF

    Topics Kähler geometry · Same authors Pak-Yeung Chan, Man-Chun Lee

  7. Ricci flow and PIC1

    Peter M. Topping PDF

    Topics Sphere theorems · Shared terms “PIC”

  8. Rigidity of area non-increasing maps

    Man-Chun Lee, Luen-Fai Tam and Jingbo Wan PDF

    Topics Sphere theorems · Same author Man-Chun Lee

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) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.