Paper · Sep 10, 2017

Yamabe Flow: steady solitons and type II singularities

Beomjun Choi, Panagiota Daskalopoulos

math.DGarXiv:1709.03192
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Abstract

We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time or as .

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Submitted
Sep 10, 2017
Subjects
math.DG, math.AP
Yamabe flowRicci solitonsSingularitiesBoundaries & non-compact

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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 New type I ancient compact solutions of the Yamabe flow (Panagiota Daskalopoulos, Manuel del Pino, John King et al., 2016)2Daskalopoulo 2016 Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds (Panagiota Daskalopoulos, Yannick Sire and Juan-Luis Vázquez, 2017)5Daskalopoulo 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)4Ho 2017 Type II Singularities on complete non-compact Yamabe flow (Beomjun Choi, Panagiota Daskalopoulos and John King, 2018)1Choi 2018 CR Yamabe constant, CR Yamabe flow and its soliton (Pak Tung Ho and Kunbo Wang, 2020)8Ho 2020 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)6Takahashi 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)7Carron 2022 On the existence and classification of k-Yamabe gradient solitons (Maria Fernanda Espinal and Mariel Sáez, 2024)3Espinal 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type II Singularities on complete non-compact Yamabe flow

    Beomjun Choi, Panagiota Daskalopoulos and John King

    Topics Yamabe flow, Boundaries & non-compact, Singularities · Same authors Beomjun Choi, Panagiota Daskalopoulos · Shared terms “steady soliton”, “compact conformally”, “conformally flat”

  2. New type I ancient compact solutions of the Yamabe flow

    Panagiota Daskalopoulos, Manuel del Pino, John King et al.Wider flows

    Topics Yamabe flow, Boundaries & non-compact, Singularities · Same author Panagiota Daskalopoulos

  3. On the existence and classification of -Yamabe gradient solitons

    Maria Fernanda Espinal and Mariel Sáez

    Topics Yamabe flow, Ricci solitons · Shared terms “conformally flat”

  4. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  5. Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds

    Panagiota Daskalopoulos, Yannick Sire and Juan-Luis VázquezWider flows

    Topics Yamabe flow · Same author Panagiota Daskalopoulos · Shared terms “conformally flat”

  6. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Yamabe flow, Boundaries & non-compact, Singularities

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

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Yamabe flow, Boundaries & non-compact

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

    Pak Tung Ho and Kunbo WangWider flows

    Topics Yamabe flow, Boundaries & non-compact

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) 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)

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.