Paper · Apr 22, 2016

Convergence of the Yamabe flow on manifolds with minimal boundary

Sergio Almaraz, Liming Sun

math.DGWider flowsarXiv:1604.06789v2
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Abstract

We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.

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Submitted
Apr 22, 2016
Last revised
Dec 26, 2018 (v2)
Subjects
math.DG
DOI
10.2422/2036-2145.201608_001

Comments: Final version. Accepted to Annali della Scuola Normale Superiore di Pisa (Classe di Scienze). arXiv admin note: text overlap with arXiv:1206.1184"

Yamabe flowScalar curvature

Its neighbourhood

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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. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  2. Yamabe Flow with prescribed scalar curvature

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    Topics Yamabe flow, Scalar curvature · Shared terms “compact Riemannian”, “Riemannian manifold”

  3. Convergence of the fractional Yamabe flow for a class of initial data

    Hardy Chan, Yannick Sire and Liming SunWider flows

    Topics Yamabe flow · Same author Liming Sun

  4. The Exponential Convergence of the CR Yamabe Flow

    Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow, Scalar curvature

  5. First eigenvalues of geometric operators under the Yamabe flow

    Pak Tung HoWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “compact Riemannian”, “Riemannian manifold”

  6. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

    Topics Yamabe flow, Scalar curvature

  7. On the Chern-Yamabe flow

    Mehdi Lejmi and Ali MaalaouiWider flows

    Topics Yamabe flow, Scalar curvature

  8. Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability

    Li MaWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “Riemannian manifold”

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.

Foundational paper in the Yamabe flow guide: Almaraz–Sun: convergence on manifolds with minimal boundary.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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.