Paper · Jun 21, 2017

On the uniqueness of Ricci flow

Man-Chun Lee

math.DGarXiv:1706.06848v3
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Abstract

In this note, we study the problem of uniqueness of Ricci flow on complete noncompact manifolds. We consider the class of solutions with curvature bounded above by C/t when t > 0. In paricular, we proved uniqueness if in addition the initial curvature is of polynomial growth and Ricci curvature of the flow is relatively small.

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Submitted
Jun 21, 2017
Last revised
Oct 20, 2018 (v3)
Subjects
math.DG

Comments: All comments are welcome!

Ricci flowStability & uniquenessBoundaries & 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 On existence and curvature estimates of Ricci flow (Man-Chun Lee and Luen-Fai Tam, 2017)6Lee 2017 Chern-Ricci flows on noncompact complex manifolds (Man-Chun Lee and Luen-Fai Tam, 2017)4Lee 2017 Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures (Man-Chun Lee and John Man-shun Ma, 2017)3Lee 2017 Second Ricci flow on noncompact Hermitian manifolds (Man-Chun Lee, 2018)7Lee 2018 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)5Lee 2019 Small curvature concentration and Ricci flow smoothing (Pak-Yeung Chan, Eric Chen and Man-Chun Lee, 2021)2Chan 2021 Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds (Liang Cheng and Yongjia Zhang, 2021)1Cheng 2021 Uniqueness of Ricci flow with scaling invariant estimates (Man-Chun Lee, 2025)8Lee 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds

    Liang Cheng and Yongjia Zhang

    Topics Boundaries & non-compact, Stability & uniqueness · Shared terms “noncompact manifolds”, “complete noncompact”

  2. Small curvature concentration and Ricci flow smoothing

    Pak-Yeung Chan, Eric Chen and Man-Chun Lee

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “noncompact manifolds”, “complete noncompact”

  3. Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures

    Man-Chun Lee and John Man-shun Ma

    Topics Boundaries & non-compact, Stability & uniqueness · Same author Man-Chun Lee

  4. Chern-Ricci flows on noncompact complex manifolds

    Man-Chun Lee and Luen-Fai Tam

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “noncompact manifolds”, “complete noncompact”

  5. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “curvature bounded”, “complete noncompact”

  6. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “complete noncompact”

  7. Second Ricci flow on noncompact Hermitian manifolds

    Man-Chun Lee

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “complete noncompact”

  8. Uniqueness of Ricci flow with scaling invariant estimates

    Man-Chun Lee

    Topics Boundaries & non-compact, Stability & uniqueness · Same author Man-Chun Lee

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.