Paper · Nov 21, 2018

Unconditional existence of conformally hyperbolic Yamabe flows

Mario B. Schulz

math.APWider flowsarXiv:1811.08798v2
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As on arXiv

Abstract

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.

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Submitted
Nov 21, 2018
Last revised
Jul 3, 2019 (v2)
Subjects
math.AP, math.DG
Journal
Analysis & PDE 13 (2020) 1579-1590
DOI
10.2140/apde.2020.13.1579
Yamabe flowStability & uniquenessLong-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 Instantaneously complete Yamabe flow on hyperbolic space (Mario B. Schulz, 2016)1Schulz 2016 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)5Ho 2017 Yamabe flow on non-compact manifolds with unbounded initial curvature (Mario B. Schulz, 2018)2Schulz 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)7Ma 2020 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)6Takahashi 2021 Combinatorial Yamabe flow on hyperbolic bordered surfaces (Shengyu Li, Xu Xu and Ze Zhou, 2022)8Li 2022 Some properties of hyperbolic Yamabe solitons (Adara M. Blaga and Cihan Özgür, 2023)4Blaga 2023
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, 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, Long-time existence & convergence · Same author Mario B. Schulz · Shared terms “require initial”, “conformally”, “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. Some properties of hyperbolic Yamabe solitons

    Adara M. Blaga and Cihan ÖzgürWider flows

    Topics Yamabe flow · Shared terms “hyperbolic Yamabe”

  5. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  6. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

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

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

    Li MaWider flows

    Topics Yamabe flow, Stability & uniqueness · Shared terms “complete Yamabe”, “initial metric”

  8. Combinatorial Yamabe flow on hyperbolic bordered surfaces

    Shengyu Li, Xu Xu and Ze ZhouWider flows

    Topics Yamabe flow, 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.