Paper · Dec 8, 2016

Instantaneously complete Yamabe flow on hyperbolic space

Mario B. Schulz

math.APWider flowsarXiv:1612.02745v4
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Abstract

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension . The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.

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Submitted
Dec 8, 2016
Last revised
Oct 31, 2019 (v4)
Subjects
math.AP
Journal
Calc. Var. (2019) 58: 190
DOI
10.1007/s00526-019-1634-9
Yamabe flowScalar curvatureLong-time existence & convergenceRotational symmetry & warped products

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 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)5Amacha 2017 Yamabe flow on non-compact manifolds with unbounded initial curvature (Mario B. Schulz, 2018)2Schulz 2018 The Exponential Convergence of the CR Yamabe Flow (Weimin Sheng and Kunbo Wang, 2018)7Sheng 2018 Unconditional existence of conformally hyperbolic Yamabe flows (Mario B. Schulz, 2018)1Schulz 2018 Incomplete Yamabe flows and removable singularities (Mario B. Schulz, 2019)3Schulz 2019 Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability (Li Ma, 2020)4Ma 2020 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)6Lye 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “conformally hyperbolic”, “complete Yamabe”, “instantaneously complete”

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

    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”

  3. Incomplete Yamabe flows and removable singularities

    Mario B. SchulzWider flows

    Topics Yamabe flow · Same author Mario B. Schulz · Shared terms “instantaneously complete”

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

    Li MaWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “complete Yamabe”, “initial metric”

  5. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

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

  6. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

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

  7. The Exponential Convergence of the CR Yamabe Flow

    Weimin Sheng and Kunbo WangWider 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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.