Paper · Apr 27, 2016

Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature

Fei He

math.DGarXiv:1604.07923
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Abstract

We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions on noncompact Riemannian manifolds with at most quadratic curvature growth, whose Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method implies that, under some appropriate assumptions, the spatial transfer rate (the rate at which high curvature regions affect low curvature regions) of the Ricci flow resembles that of the heat equation.

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Submitted
Apr 27, 2016
Subjects
math.DG

Comments: 36 pages

Ricci flow4-manifoldsBoundaries & non-compactHarmonic 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, Boundaries & non-compact · Same author Fei He · Shared terms “unbounded curvature”, “short time”, “quadratic”

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

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “short time”, “time existence”, “heat equation”

  3. Conformal Ricci flow on asymptotically hyperbolic manifolds

    Peng Lu, Jie Qing and Yu Zheng

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “short time”, “time existence”

  4. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon

    Topics Harmonic maps & DeTurck trick, 4-manifolds, Boundaries & non-compact

  5. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

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

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

    Raphael Hochard

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

  7. 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”

  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, 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.

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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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) 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. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.