Paper · Dec 26, 2018

Local derivative estimates for the heat equation coupled to the Ricci flow

Hong Huang

math.DGarXiv:1812.10296v8
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Abstract

In this note we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang's work, we extend some results of Zhang and Bamler-Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact manifolds to the noncompact case.

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Submitted
Dec 26, 2018
Last revised
Mar 1, 2021 (v8)
Subjects
math.DG, math.AP

Comments: 31 pages, added an appendix; to appear in Communications in Contemporary Mathematics

Ricci flowBoundaries & non-compact

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Conformal Ricci flow on asymptotically hyperbolic manifolds (Peng Lu, Jie Qing and Yu Zheng, 2018)2Lu 2018 Notes on Ricci flows with collapsing initial data (I): Distance distortion (Shaosai Huang, 2018)6Huang 2018 Compact manifolds of dimension n≥ 12 with positive isotropic curvature (Hong Huang, 2019)4Huang 2019 Time analyticity for heat equation on gradient shrinking Ricci solitons (Jia-Yong Wu, 2019)7Wu 2019 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)1Huang 2021 Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition (Ling Wu, 2022)8Wu 2022 Classification of compact manifolds with positive isotropic curvature (Hong Huang, 2023)5Huang 2023 Open manifolds with uniformly positive isotropic curvature (Hong Huang, 2023)3Huang 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang

    Topics Ricci flow · Same author Hong Huang · Shared terms “equation coupled”, “heat equation”

  2. Conformal Ricci flow on asymptotically hyperbolic manifolds

    Peng Lu, Jie Qing and Yu Zheng

    Topics Boundaries & non-compact · Shared terms “derivative estimates”, “Shi type”

  3. Open manifolds with uniformly positive isotropic curvature

    Hong Huang

    Topics Boundaries & non-compact · Same author Hong Huang

  4. Compact manifolds of dimension with positive isotropic curvature

    Hong Huang

    Topics Ricci flow · Same author Hong Huang · Shared terms “compact manifolds”

  5. Classification of compact manifolds with positive isotropic curvature

    Hong Huang

    Topics Ricci flow · Same author Hong Huang · Shared terms “compact manifolds”

  6. Notes on Ricci flows with collapsing initial data (I): Distance distortion

    Shaosai Huang

    Topics Ricci flow · Shared terms “distance distortion”, “distortion estimate”

  7. Time analyticity for heat equation on gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Boundaries & non-compact · Shared terms “heat equation”, “backward”

  8. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu

    Topics Boundaries & non-compact · Shared terms “heat equation”, “Zhang”, “backward”

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) 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)

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.