Paper · Jul 3, 2019

Incomplete Yamabe flows and removable singularities

Mario B. Schulz

math.DGWider flowsarXiv:1907.02059v2
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Abstract

We study the Yamabe flow on a Riemannian manifold of dimension minus a closed submanifold of dimension and prove that there exists an instantaneously complete solution if and only if . In the remaining cases including the borderline case, we show that the removability of the -dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.

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Submitted
Jul 3, 2019
Last revised
Aug 5, 2019 (v2)
Subjects
math.DG, math.AP
DOI
10.1016/j.jfa.2020.108475
Yamabe flowSingularitiesSurfaces

Its neighbourhood

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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 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)6Bahuaud 2016 Instantaneously complete Yamabe flow on hyperbolic space (Mario B. Schulz, 2016)1Schulz 2016 Yamabe Flow: steady solitons and type II singularities (Beomjun Choi and Panagiota Daskalopoulos, 2017)7Choi 2017 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)3Schulz 2018 Unconditional existence of conformally hyperbolic Yamabe flows (Mario B. Schulz, 2018)2Schulz 2018 Yamabe flow on modified Riemann extension (Harish D, 2019)8D 2019 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)4Takahashi 2021
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

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  2. Unconditional existence of conformally hyperbolic Yamabe flows

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

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  4. Infinite-time incompleteness of noncompact Yamabe flow

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    Topics Yamabe flow, Singularities · Shared terms “incomplete”

  5. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  6. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

    Topics Yamabe flow, Singularities · Shared terms “incomplete”, “Riemannian manifold”

  7. Yamabe Flow: steady solitons and type II singularities

    Beomjun Choi and Panagiota Daskalopoulos

    Topics Yamabe flow, Singularities

  8. Yamabe flow on modified Riemann extension

    Harish DWider flows

    Topics Yamabe flow

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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.