Paper · Jun 20, 2019

A fractional conformal curvature flow on the unit sphere

Xuezhang Chen, Pak Tung Ho

math.DGWider flowsarXiv:1906.08434v3
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Abstract

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent . This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.

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Submitted
Jun 20, 2019
Last revised
Nov 20, 2021 (v3)
Subjects
math.DG, math.AP

Comments: An intermediate version, still needs to be updated

Other curvature flowsScalar curvature

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 Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant (Xuezhang Chen, Pak Tung Ho and Liming Sun, 2016)1Chen 2016 First eigenvalues of geometric operators under the Yamabe flow (Pak Tung Ho, 2018)7Ho 2018 On eternal mean curvature flows of tori in perturbations of the unit sphere (Claudia V. S. Magaño and Graham Smith, 2020)5Magaño 2020 Prescribed Webster scalar curvature flow on CR manifold (Pak Tung Ho, Quoc Anh Ngo and Hong Zhang, 2021)2Ho 2021 Convergence rate of the prescribed curvature flow (Pak Tung Ho and Jinwoo Shin, 2021)3Ho 2021 Convergence rate of the Q-curvature flow (Pak Tung Ho and Sanghoon Lee, 2024)8Ho 2024 Convergence of the fractional Yamabe flow for arbitrary initial energy (Jingeon An-Lacroix, Hardy Chan and Pak Tung Ho, 2025)4An-Lacroix 2025 Prescribed T-curvature flow on the four-dimensional unit ball (Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun et al., 2026)6Ho 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

    Xuezhang Chen, Pak Tung Ho and Liming SunWider flows

    Topics Scalar curvature · Same authors Xuezhang Chen, Pak Tung Ho

  2. Prescribed Webster scalar curvature flow on CR manifold

    Pak Tung Ho, Quoc Anh Ngo and Hong ZhangWider flows

    Topics Scalar curvature, Other curvature flows · Same author Pak Tung Ho

  3. Convergence rate of the prescribed curvature flow

    Pak Tung Ho and Jinwoo ShinWider flows

    Topics Scalar curvature, Other curvature flows · Same author Pak Tung Ho

  4. Convergence of the fractional Yamabe flow for arbitrary initial energy

    Jingeon An-Lacroix, Hardy Chan and Pak Tung HoWider flows

    Same author Pak Tung Ho · Shared terms “fractional”, “Invent”, “Math”

  5. On eternal mean curvature flows of tori in perturbations of the unit sphere

    Claudia V. S. Magaño and Graham SmithWider flows

    Shared terms “unit sphere”, “standard unit”, “perturbation”

  6. Prescribed -curvature flow on the four-dimensional unit ball

    Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun et al.Wider flows

    Topics Other curvature flows · Same author Pak Tung Ho

  7. First eigenvalues of geometric operators under the Yamabe flow

    Pak Tung HoWider flows

    Topics Scalar curvature · Same author Pak Tung Ho

  8. Convergence rate of the -curvature flow

    Pak Tung Ho and Sanghoon LeeWider flows

    Topics Other curvature flows · Same author Pak Tung Ho

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.