Paper · Jun 15, 2018

Yamabe flow on non-compact manifolds with unbounded initial curvature

Mario B. Schulz

math.APWider flowsarXiv:1806.06021v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove global existence of Yamabe flows on non-compact manifolds of dimension under the assumption that the initial metric is conformally equivalent to a complete background metric of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of can be unbounded from above and below without growth condition.

View PDF View HTML arXiv page
Submitted
Jun 15, 2018
Last revised
Nov 19, 2018 (v2)
Subjects
math.AP, math.DG
DOI
10.1007/s12220-019-00238-8
Yamabe flowScalar curvatureBoundaries & non-compactLong-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 Convergence of the Yamabe flow on manifolds with minimal boundary (Sergio Almaraz and Liming Sun, 2016)8Almaraz 2016 Instantaneously complete Yamabe flow on hyperbolic space (Mario B. Schulz, 2016)1Schulz 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)3Amacha 2017 The Exponential Convergence of the CR Yamabe Flow (Weimin Sheng and Kunbo Wang, 2018)4Sheng 2018 Unconditional existence of conformally hyperbolic Yamabe flows (Mario B. Schulz, 2018)2Schulz 2018 Global Yamabe flow on asymptotically flat manifolds (Li Ma, 2021)6Ma 2021 Normalized Yamabe flow on manifolds with bounded geometry (Bruno Caldeira, Luiz Hartmann and Boris Vertman, 2021)5Caldeira 2021 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)7Lye 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Instantaneously complete Yamabe flow on hyperbolic space

    Mario B. SchulzWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Same author Mario B. Schulz · Shared terms “require initial”, “conformal factor”, “global existence”

  2. Unconditional existence of conformally hyperbolic Yamabe flows

    Mario B. SchulzWider flows

    Topics Yamabe flow, Long-time existence & convergence · Same author Mario B. Schulz · Shared terms “require initial”, “conformally”, “global existence”

  3. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

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

  4. The Exponential Convergence of the CR Yamabe Flow

    Weimin Sheng and Kunbo WangWider flows

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

  5. 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

  6. Global Yamabe flow on asymptotically flat manifolds

    Li MaWider flows

    Topics Yamabe flow, Scalar curvature, Boundaries & non-compact · Shared terms “background metric”, “initial metric”

  7. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

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

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

    Sergio Almaraz and Liming SunWider flows

    Topics Yamabe flow, Scalar curvature

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.