Paper · Dec 15, 2021

Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds

Liang Cheng, Yongjia Zhang

math.DGarXiv:2112.07818v2
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Abstract

In this paper, we prove a pseudolocality-type theorem for -complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the -complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L. Chen.

Ricci flowStability & uniquenessBoundaries & non-compact

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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 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 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)2Cheng 2021 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)6Huang 2021 Pseudolocality theorems of Ricci flows on incomplete manifolds (Liang Cheng, 2022)1Cheng 2022 Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows (Albert Chau and Adam Martens, 2023)8Chau 2023 An ε-regularity theorem for Perelman's reduced volume (Liang Cheng and Yongjia Zhang, 2025)4Cheng 2025 Rigidity and gap theorems for Ricci shrinkers (Pak-Yeung Chan and Yongjia Zhang, 2026)7Chan 2026 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)5Huang 2026 Pseudolocality and strong uniqueness for complete Ricci flows (Shaochuang Huang and Zhuo Peng, 2026)3Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pseudolocality theorems of Ricci flows on incomplete manifolds

    Liang Cheng PDF

    Topics Boundaries & non-compact · Same author Liang Cheng · Shared terms “almost Euclidean”, “pseudolocality”, “Euclidean space”

  2. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

    Topics Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang · Shared terms “Euclidean space”

  3. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Stability & uniqueness · Shared terms “pseudolocality”, “complete Ricci”, “strong uniqueness”

  4. An -regularity theorem for Perelman's reduced volume

    Liang Cheng and Yongjia Zhang PDF

    Topics Ricci flow · Same authors Liang Cheng, Yongjia Zhang

  5. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics Boundaries & non-compact, Stability & uniqueness · Shared terms “strong uniqueness”, “Euclidean”

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

    Shaochuang Huang and Luen-Fai Tam PDF

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

  7. Rigidity and gap theorems for Ricci shrinkers

    Pak-Yeung Chan and Yongjia Zhang PDF

    Topics Stability & uniqueness · Same author Yongjia Zhang

  8. Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

    Albert Chau and Adam Martens PDF

    Topics Boundaries & non-compact · Shared terms “pseudolocality”, “sectional curvature”, “complete noncompact”

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.