Paper · Apr 7, 2017

Another proof of the local curvature estimate for the Ricci flow

Shu-Yu Hsu

math.DGarXiv:1704.02209v3
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Abstract

By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature.

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Submitted
Apr 7, 2017
Last revised
Jan 18, 2018 (v3)
Subjects
math.DG, math.AP

Comments: 11 pages, typos corrected

Ricci flow

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  7. Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

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    Topics Ricci flow · Shared terms “curvature estimates”, “uniform bounds”, “curvature tensor”

  8. Optimal curvature estimates for homogeneous Ricci flows

    Christoph Böhm, Ramiro Lafuente and Miles Simon

    Topics Ricci flow · Shared terms “curvature estimates”, “curvature tensor”

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.