Paper · Sep 14, 2018

Type II Singularities on complete non-compact Yamabe flow

Beomjun Choi, Panagiota Daskalopoulos, John King

math.DGarXiv:1809.05281
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Abstract

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

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Submitted
Sep 14, 2018
Subjects
math.DG

Comments: 36 pages, 2 figures

Yamabe flowRicci solitonsSingularitiesBoundaries & non-compactRotational 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 New type I ancient compact solutions of the Yamabe flow (Panagiota Daskalopoulos, Manuel del Pino, John King et al., 2016)2Daskalopoulo 2016 Instantaneously complete Yamabe flow on hyperbolic space (Mario B. Schulz, 2016)8Schulz 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)4Daskalopoulo 2017 Yamabe Flow: steady solitons and type II singularities (Beomjun Choi and Panagiota Daskalopoulos, 2017)1Choi 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)5Ho 2017 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)7Choi 2018 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)6Takahashi 2021 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. Yamabe Flow: steady solitons and type II singularities

    Beomjun Choi and Panagiota Daskalopoulos

    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, Rotational symmetry & warped products, Boundaries & non-compact · Same authors Panagiota Daskalopoulos, John King · Shared terms “rotationally symmetric”

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

    Maria Fernanda Espinal and Mariel Sáez

    Topics Yamabe flow, Rotational symmetry & warped products, Ricci solitons · Shared terms “conformally flat”, “rotationally symmetric”

  4. 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”

  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, Boundaries & non-compact, Singularities

  7. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Boundaries & non-compact · Same authors Beomjun Choi, Panagiota Daskalopoulos

  8. Instantaneously complete Yamabe flow on hyperbolic space

    Mario B. SchulzWider flows

    Topics Yamabe flow, Rotational symmetry & warped products · Shared terms “rotationally symmetric”, “conformally”

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.