Paper · Nov 8, 2018

Weakly PIC1 manifolds with maximal volume growth

Fei He, Man-Chun Lee

math.DGarXiv:1811.03318
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Abstract

In this article we use Ricci flow to show that complete PIC1 manifolds with maximal volume growth are diffeomorphic to . One of the key ingredients is local estimates of curvature lower bounds on an initial time interval of the Ricci flow. As another application of these estimates we obtain pseudolocality type results related to the PIC1 condition.

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Submitted
Nov 8, 2018
Subjects
math.DG

Comments: 18 pages

Ricci flowSphere theorems

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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 Chern-Ricci flows on noncompact complex manifolds (Man-Chun Lee and Luen-Fai Tam, 2017)8Lee 2017 Shrinking Ricci solitons with positive isotropic curvature (Keaton Naff, 2019)6Naff 2019 Ancient solutions to the Ricci flow with isotropic curvature conditions (Jae Ho Cho and Yu Li, 2020)7Cho 2020 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)2Lee 2022 Ricci flow and PIC1 (Peter M. Topping, 2023)4Topping 2023 Rigidity of area non-increasing maps (Man-Chun Lee, Luen-Fai Tam and Jingbo Wan, 2023)3Lee 2023 Expanding Ricci solitons coming out of weakly PIC1 metric cones (Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey, 2024)1Chan 2024 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)5Deruelle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Expanding Ricci solitons coming out of weakly PIC1 metric cones

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

    Topics Sphere theorems · Same author Man-Chun Lee · Shared terms “weakly PIC”, “volume growth”

  2. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics Sphere theorems · Same author Man-Chun Lee · Shared terms “PIC”

  3. Rigidity of area non-increasing maps

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

    Topics Sphere theorems · Same author Man-Chun Lee

  4. Ricci flow and PIC1

    Peter M. Topping

    Topics Sphere theorems · Shared terms “PIC”

  5. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al.

    Topics Sphere theorems · Same author Man-Chun Lee · Shared terms “PIC”

  6. Shrinking Ricci solitons with positive isotropic curvature

    Keaton Naff

    Topics Sphere theorems · Shared terms “PIC”

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

    Jae Ho Cho and Yu Li

    Topics Sphere theorems · Shared terms “weakly PIC”

  8. Chern-Ricci flows on noncompact complex manifolds

    Man-Chun Lee and Luen-Fai Tam

    Topics Ricci flow · Same author Man-Chun Lee · Shared terms “maximal volume”, “volume growth”

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) 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)

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. 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.