Paper · May 19, 2017

Yamabe Flow with prescribed scalar curvature

Inas Amacha, Rachid Regbaoui

math.DGWider flowsarXiv:1705.06898
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Abstract

In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.

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Submitted
May 19, 2017
Subjects
math.DG
Journal
Pacific J. Math. 297 (2018) 257-275
DOI
10.2140/pjm.2018.297.257
Yamabe flowScalar curvatureLong-time existence & convergence

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.

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 Convergence of the Yamabe flow on manifolds with minimal boundary (Sergio Almaraz and Liming Sun, 2016)4Almaraz 2016 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)3Bahuaud 2016 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)8Ho 2017 The Exponential Convergence of the CR Yamabe Flow (Weimin Sheng and Kunbo Wang, 2018)5Sheng 2018 Normalized Yamabe flow on manifolds with bounded geometry (Bruno Caldeira, Luiz Hartmann and Boris Vertman, 2021)6Caldeira 2021 Convergence rate of the prescribed curvature flow (Pak Tung Ho and Jinwoo Shin, 2021)1Ho 2021 Convergence of the Weighted Yamabe Flow (Zetian Yan, 2021)7Yan 2021 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)2Lye 2024
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  1. Convergence rate of the prescribed curvature flow

    Pak Tung Ho and Jinwoo ShinWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “prescribed scalar”

  2. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Shared terms “negative scalar”

  3. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Shared terms “negative scalar”, “compact Riemannian”, “Riemannian manifold”

  4. Convergence of the Yamabe flow on manifolds with minimal boundary

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

  5. The Exponential Convergence of the CR Yamabe Flow

    Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence

  6. Normalized Yamabe flow on manifolds with bounded geometry

    Bruno Caldeira, Luiz Hartmann and Boris VertmanWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Shared terms “negative scalar”, “Riemannian manifold”

  7. Convergence of the Weighted Yamabe Flow

    Zetian YanWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence

  8. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

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.

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.