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Book 11A

The Ricci Flow: Existence and Maximum Principles

Hamilton’s flow, part one

The equation and its first solutions, the evolution of curvature, short-time existence by DeTurck’s trick, maximum principles, pinching, Hamilton’s 1982 theorem, surfaces and homogeneous flows.

  • Main companionLectures on the Ricci Flow · Peter Topping
  • Main companionThe Ricci Flow: An Introduction · Chow & Knopf
In preparation

Contents

  1. 1
    The Equation and Its First Solutions

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

    In preparation

  2. 2
    How Curvature Evolves

    The variation formulas and the evolution equations, computed once and carefully.

    In preparation

  3. 3
    Short-Time Existence and Uniqueness

    Why the flow is only weakly parabolic, DeTurck’s trick, and Shi’s estimates.

    In preparation

  4. 4
    Maximum Principles under Ricci Flow

    Scalar and tensor maximum principles, and preserved curvature conditions.

    In preparation

  5. 5
    Hamilton–Ivey Pinching

    Why high curvature in dimension 3 is almost positive.

    In preparation

  6. 6
    Hamilton’s 1982 Theorem

    Positive Ricci curvature in dimension 3 flows to a round metric.

    In preparation

  7. 7
    Ricci Flow on Surfaces

    Uniformization by Ricci flow, and Hamilton’s entropy.

    In preparation

  8. 8
    Homogeneous Flows

    A computable laboratory: the flow on Thurston’s geometries.

    In preparation

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