Paper · Mar 26, 2025

Uniqueness of Ricci flow with scaling invariant estimates

Man-Chun Lee

math.DGarXiv:2503.20292v2
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Abstract

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

Comments: 24 pages. Printing mistakes corrected. Minor detail added

Ricci flow(Non)collapsing3-manifoldsStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trick

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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 2021, 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 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)2Huang 2021 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)6Lee 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)3Lee 2022 Ricci Flow under Kato-type curvature lower bound (Man-Chun Lee, 2023)5Lee 2023 Gap Theorem on Riemannian manifolds using Ricci flow (Pak-Yeung Chan and Man-Chun Lee, 2023)1Chan 2023 Rotationally symmetric Ricci Flow on ℝⁿ⁺¹ (Ming Hsiao, 2025)7Hsiao 2025 Higher-dimensional flying wing Steady Ricci Solitons (Pak-Yeung Chan, Yi Lai and Man-Chun Lee, 2025)4Chan 2025 On the uniqueness of Ricci flows from certain Alexandrov spaces (Laura Flower, 2026)8Flower 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gap Theorem on Riemannian manifolds using Ricci flow

    Pak-Yeung Chan and Man-Chun Lee PDF

    Topics (Non)collapsing, 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “scaling invariant”, “invariant curvature”

  2. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Stability & uniqueness · Shared terms “map heat”, “heat flow”, “Chen Zhu”

  3. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “map heat”, “heat flow”

  4. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee PDF

    Topics (Non)collapsing, Boundaries & non-compact, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “invariant estimate”, “scaling invariant”, “collapsed”

  5. Ricci Flow under Kato-type curvature lower bound

    Man-Chun Lee PDF

    Topics (Non)collapsing, 3-manifolds, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “collapsed”

  6. Three-manifolds with non-negatively pinched Ricci curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “negatively”, “invariant curvature”, “complete Ricci”

  7. Rotationally symmetric Ricci Flow on

    Ming Hsiao PDF

    Topics Harmonic maps & DeTurck trick, (Non)collapsing · Shared terms “scaling invariant”, “complete Ricci”, “invariant curvature”

  8. On the uniqueness of Ricci flows from certain Alexandrov spaces

    Laura Flower PDF

    Topics (Non)collapsing, Boundaries & non-compact, Stability & uniqueness · Shared terms “scaling invariant”, “invariant curvature”, “curvature bounds”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.