Paper · May 29, 2021

Normalized Yamabe flow on manifolds with bounded geometry

Bruno Caldeira, Luiz Hartmann, Boris Vertman

math.DGWider flowsarXiv:2105.14282v3
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Abstract

The goal of this paper is to study Yamabe flow on a complete Riemannian manifold of bounded geometry with possibly infinite volume. In the case of infinite volume, standard volume normalization of the Yamabe flow fails and the flow may not converge. Instead, we consider a curvature normalized Yamabe flow, and assuming negative scalar curvature, prove its long-time existence and convergence. This extends the results of Suárez-Serrato and Tapie to a non-compact setting. In the appendix, we specify our analysis of a particular example of manifolds with bounded geometry, namely manifolds with fibered boundary metric. In this case, we obtain stronger estimates for the short-time solution using microlocal methods.

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Submitted
May 29, 2021
Last revised
Oct 13, 2022 (v3)
Subjects
math.DG, math.AP

Comments: 37 pages, 1 figure; v2: typos and reference corrected, v3: generalization to bounded geometry

Yamabe flowScalar curvatureBoundaries & non-compactLong-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 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)1Bahuaud 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)6Amacha 2017 Yamabe flow on non-compact manifolds with unbounded initial curvature (Mario B. Schulz, 2018)7Schulz 2018 The Exponential Convergence of the CR Yamabe Flow (Weimin Sheng and Kunbo Wang, 2018)8Sheng 2018 Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant (Jørgen Olsen Lye and Boris Vertman, 2020)3Lye 2020 Convergence of the Yamabe flow on singular spaces with positive Yamabe constant (Gilles Carron, Jørgen Olsen Lye and Boris Vertman, 2021)5Carron 2021 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)4Takahashi 2021 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)2Lye 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

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

  2. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

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

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

    Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Same author Boris Vertman · Shared terms “normalized Yamabe”

  4. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Yamabe flow, Long-time existence & convergence, Boundaries & non-compact · Shared terms “infinite”

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

    Gilles Carron, Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Yamabe flow · Same author Boris Vertman · Shared terms “normalized Yamabe”

  6. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

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

  7. Yamabe flow on non-compact manifolds with unbounded initial curvature

    Mario B. SchulzWider flows

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

  8. The Exponential Convergence of the CR Yamabe Flow

    Weimin Sheng and Kunbo WangWider flows

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

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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.