Paper · Mar 29, 2016

Short-time existence of the Ricci flow on complete, non-collapsed -manifolds with Ricci curvature bounded from below

Raphael Hochard

math.DGarXiv:1603.08726
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Abstract

We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.

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Submitted
Mar 29, 2016
Subjects
math.DG

Comments: 39 pages, 6 figures

Ricci flow(Non)collapsing3-manifoldsHarmonic maps & DeTurck trick

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence and applications of Ricci flows via pseudolocality

    Fei He

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

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

    Kaveh Eftekharinasab

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

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

    Shu-Cheng Chang and Chin-Tung Wu

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

  4. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

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

  5. Uniqueness of Ricci flow with scaling invariant estimates

    Man-Chun Lee

    Topics Harmonic maps & DeTurck trick, (Non)collapsing, 3-manifolds · Shared terms “collapsed”, “curvature bounds”

  6. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

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

  7. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, (Non)collapsing · Shared terms “short time”, “time existence”

  8. The Ricci flow under almost non-negative curvature conditions

    Richard H. Bamler, Esther Cabezas-Rivas and Burkhard Wilking

    Topics Harmonic maps & DeTurck trick, (Non)collapsing · Shared terms “short time”, “curvature bounds”, “collapsed”

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.

Foundational paper in the Boundaries & non-compact guide: Hochard: short-time existence on complete non-collapsed 3-manifolds with Ricci curvature bounded below.

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.