Paper · Jun 2, 2020

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

Jørgen Olsen Lye, Boris Vertman

math.APWider flowsarXiv:2006.01544
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Abstract

In this work we establish long-time existence of the normalized Yamabe flow with positive Yamabe constant on a class of manifolds that includes spaces with incomplete cone-edge singularities. We formulate our results axiomatically, so that our results extend to general stratified spaces as well, provided certain parabolic Schauder estimates hold. The central analytic tool is a parabolic Moser iteration, which yields uniform upper and lower bounds on both the solution and the scalar curvature.

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Submitted
Jun 2, 2020
Subjects
math.AP, math.DG
Journal
Analysis & PDE 16 (2023) 477-510
DOI
10.2140/apde.2023.16.477

Comments: 36 pages,

Yamabe flowSingularitiesScalar curvatureWeak & singular flowsLong-time existence & convergence

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 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)3Bahuaud 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)6Amacha 2017 CR Yamabe constant, CR Yamabe flow and its soliton (Pak Tung Ho and Kunbo Wang, 2020)7Ho 2020 Normalized Yamabe flow on manifolds with bounded geometry (Bruno Caldeira, Luiz Hartmann and Boris Vertman, 2021)4Caldeira 2021 Convergence of the Yamabe flow on singular spaces with positive Yamabe constant (Gilles Carron, Jørgen Olsen Lye and Boris Vertman, 2021)1Carron 2021 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)8Takahashi 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)5Carron 2022 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. Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

    Gilles Carron, Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Yamabe flow, Weak & singular flows · Same authors Jørgen Olsen Lye, Boris Vertman · Shared terms “positive Yamabe”, “Yamabe constant”, “singular spaces”

  2. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Same authors Jørgen Olsen Lye, Boris Vertman

  3. 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 “edge singularities”, “normalized Yamabe”, “incomplete”

  4. 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 · Same author Boris Vertman · Shared terms “normalized Yamabe”

  5. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Yamabe flow · Shared terms “Yamabe constant”, “positive Yamabe”

  6. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

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

  7. CR Yamabe constant, CR Yamabe flow and its soliton

    Pak Tung Ho and Kunbo WangWider flows

    Topics Yamabe flow · Shared terms “Yamabe constant”

  8. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Yamabe flow, Long-time existence & convergence, Singularities · Shared terms “incomplete”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

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