Paper · Mar 21, 2016

Ricci de Turck flow on singular manifolds

Boris Vertman

math.DGarXiv:1603.06545v4
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Abstract

In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat incomplete edge metric, we prove long time existence of the Ricci de Turck flow. Under certain conditions, our results yield existence of Ricci flow on spaces with incomplete edge singularities. The proof works by a careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.

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Submitted
Mar 21, 2016
Last revised
Mar 2, 2020 (v4)
Subjects
math.DG, math.SP
Journal
J Geom Anal (2020)
DOI
10.1007/s12220-020-00399-x

Comments: 50 pages, 2 figures

Ricci flowSingularitiesLong-time existence & convergenceHarmonic maps & DeTurck trick

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 A simple proof of the short-time existence and uniqueness for Ricci flow (Kaveh Eftekharinasab, 2016)6Eftekharinas 2016 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)4Bahuaud 2016 Short-Time Existence Theorem for the CR Torsion Flow (Shu-Cheng Chang and Chin-Tung Wu, 2018)7Chang 2018 Perelman's Entropies for Manifolds with conical Singularities (Klaus Kroencke and Boris Vertman, 2019)2Kroencke 2019 Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci (Mahdi Ammar, 2019)8Ammar 2019 Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow (Klaus Kroencke, Tobias Marxen and Boris Vertman, 2020)1Kroencke 2020 Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant (Jørgen Olsen Lye and Boris Vertman, 2020)5Lye 2020 A Survey on the Ricci flow on Singular Spaces (Klaus Kroencke and Boris Vertman, 2021)3Kroencke 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow

    Klaus Kroencke, Tobias Marxen and Boris Vertman

    Topics Harmonic maps & DeTurck trick, Singularities · Same author Boris Vertman · Shared terms “Turck flow”

  2. Perelman's Entropies for Manifolds with conical Singularities

    Klaus Kroencke and Boris Vertman

    Topics Harmonic maps & DeTurck trick, Singularities · Same author Boris Vertman · Shared terms “Turck flow”

  3. A Survey on the Ricci flow on Singular Spaces

    Klaus Kroencke and Boris Vertman

    Topics Harmonic maps & DeTurck trick, Singularities · Same author Boris Vertman · Shared terms “Turck flow”

  4. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

    Topics Long-time existence & convergence, Singularities · Same author Boris Vertman · Shared terms “incomplete edge”, “edge singularities”, “flow starting”

  5. Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant

    Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Long-time existence & convergence, Singularities · Same author Boris Vertman · Shared terms “edge singularities”, “incomplete”

  6. A simple proof of the short-time existence and uniqueness for Ricci flow

    Kaveh Eftekharinasab

    Topics Harmonic maps & DeTurck trick · Shared terms “Riemannian metrics”, “local existence”, “short time”

  7. Short-Time Existence Theorem for the CR Torsion Flow

    Shu-Cheng Chang and Chin-Tung Wu

    Topics Harmonic maps & DeTurck trick · Shared terms “local existence”, “short time”, “time existence”

  8. Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci

    Mahdi Ammar

    Topics Harmonic maps & DeTurck trick · Shared terms “Riemannian metrics”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

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    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 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.