Paper · Jan 5, 2020

On the maximal rate of convergence under the Ricci flow

Brett Kotschwar

math.DGarXiv:2001.01272v2
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Abstract

We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.

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Submitted
Jan 5, 2020
Last revised
Sep 7, 2020 (v2)
Subjects
math.DG, math.AP

Comments: v2: Minor corrections; 20 pages, no figures

Ricci flow

Its neighbourhood

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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. On the rate of convergence of the rescaled mean curvature flow

    Rory Martin-Hagemayer and Natasa Sesum

    Shared terms “fixed exponential”, “exponential rate”

  2. Ricci flows which terminate in cones

    Brett Kotschwar

    Topics Ricci flow · Same author Brett Kotschwar

  3. The full holonomy group under the Ricci flow

    Mary Cook and Brett Kotschwar

    Topics Ricci flow · Same author Brett Kotschwar

  4. Gradient inequality and convergence of the normalized Ricci flow

    Nikos I. Kavallaris and Takashi Suzuki

    Topics Ricci flow · Shared terms “normalized Ricci”

  5. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Ricci flow · Same author Brett Kotschwar

  6. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Similar wording in the title and abstract

  7. The rate of -convergence for Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ricci flow

  8. A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

    Brett Kotschwar and Lu Wang

    Same author Brett Kotschwar

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)

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.