Paper · Jul 6, 2021

Quasiconformal Flows on non-Conformally Flat Spheres

Sun-Yung Alice Chang, Eden Prywes, Paul Yang

math.DGarXiv:2107.02785
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Abstract

We study integral curvature conditions for a Riemannian metric on that quantify the best bilipschitz constant between and the standard metric on . Our results show that the best bilipschitz constant is controlled by the -norm of the Weyl tensor and the -norm of the -curvature, under the conditions that those quantities are sufficiently small, has a positive Yamabe constant and the -curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on .

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Submitted
Jul 6, 2021
Subjects
math.DG, math.CV
Ricci flow

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some aspects of Ricci flow on the 4-sphere

    Sun-Yung Alice Chang and Eric Chen

    Topics Ricci flow · Same author Sun-Yung Alice Chang · Shared terms “Weyl tensor”, “Yamabe”

  2. A conformally invariant gap theorem characterizing via the Ricci flow

    Sun-Yung A. Chang, Matthew Gursky and Siyi Zhang

    Topics Ricci flow · Same author Sun-Yung Alice Chang · Shared terms “conformally”

  3. Evolution of relative Yamabe constant under Ricci Flow

    Boris Botvinnik and Peng Lu

    Topics Ricci flow · Shared terms “Yamabe constant”

  4. On locally conformally flat manifolds with positive pinched Ricci curvature

    Liang ChengWider flows

    Shared terms “conformally flat”, “Yamabe”

  5. Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds

    Panagiota Daskalopoulos, Yannick Sire and Juan-Luis VázquezWider flows

    Shared terms “conformally flat”, “positive Yamabe”

  6. Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

    Gilles Carron, Jørgen Olsen Lye and Boris VertmanWider flows

    Shared terms “positive Yamabe”, “Yamabe constant”

  7. Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant

    Jørgen Olsen Lye and Boris VertmanWider flows

    Shared terms “positive Yamabe”, “Yamabe constant”

  8. Gap Theorem on locally conformally flat manifold

    Ming Hsiao and Man-Chun LeeWider flows

    Shared terms “conformally flat”, “Yamabe”

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)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.