Paper · May 26, 2017

Scalar curvature flow on S^n to a prescribed sign-changing function

Hong Zhang

math.APWider flowsarXiv:1705.09627
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Abstract

In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function , which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges to a metric with the prescribed function as its scalar curvature.

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Submitted
May 26, 2017
Subjects
math.AP, math.DG

Comments: 23 pages

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 Q-curvature flow on closed manifolds of even dimension (Quôc Anh Ngô and Hong Zhang, 2017)2Ngô 2017 Asymptotic convergence for modified scalar curvature flow (Ling Xiao, 2020)8Xiao 2020 Prescribed Webster scalar curvature flow on CR manifold (Pak Tung Ho, Quoc Anh Ngo and Hong Zhang, 2021)1Ho 2021 Convergence rate of the prescribed curvature flow (Pak Tung Ho and Jinwoo Shin, 2021)4Ho 2021 A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic (Franziska Borer, Peter Elbau and Tobias Weth, 2023)6Borer 2023 Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation (Juan Carlos Fernández, Oscar Palmas and Jonatán Torres Orozco, 2023)3Fernández 2023 Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree (Weike Yu, 2025)5Yu 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)7Li 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 Hong Zhang

  2. Prescribed -curvature flow on closed manifolds of even dimension

    Quôc Anh Ngô and Hong ZhangWider flows

    Topics Other curvature flows · Same author Hong Zhang · Shared terms “prescribing”

  3. Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation

    Juan Carlos Fernández, Oscar Palmas and Jonatán Torres Orozco

    Topics Scalar curvature · Shared terms “sign changing”

  4. Convergence rate of the prescribed curvature flow

    Pak Tung Ho and Jinwoo ShinWider flows

    Topics Scalar curvature, Other curvature flows · Shared terms “prescribing scalar”

  5. Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

    Weike YuWider flows

    Topics Scalar curvature, Other curvature flows · Shared terms “candidate curvature”, “curvature function”

  6. A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

    Franziska Borer, Peter Elbau and Tobias WethWider flows

    Topics Other curvature flows · Shared terms “sign changing”, “prescribed function”

  7. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows

    Topics Scalar curvature, Other curvature flows

  8. Asymptotic convergence for modified scalar curvature flow

    Ling XiaoWider flows

    Topics Scalar curvature, Other curvature flows · Shared terms “flow converges”

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.