Paper · Sep 1, 2017

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

Man-Chun Lee, John Man-shun Ma

math.DGarXiv:1709.00253v2
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Abstract

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.

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Submitted
Sep 1, 2017
Last revised
Feb 3, 2019 (v2)
Subjects
math.DG

Comments: 25 pages; final version to appear in CAG

Ricci flowMean curvature 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 the uniqueness of Ricci flow (Man-Chun Lee, 2017)4Lee 2017 Note on the backwards uniqueness of mean curvature flow (Zhuhong Zhang, 2017)3Zhang 2017 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)6Huang 2018 Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics (Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam, 2019)5Huang 2019 Manifolds with small curvature concentration (Pak-Yeung Chan, Shaochuang Huang and Man-Chun Lee, 2022)7Chan 2022 Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds (Dasong Li and John Man Shun Ma, 2024)2Li 2024 Uniqueness of Ricci flow with scaling invariant estimates (Man-Chun Lee, 2025)1Lee 2025 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)8Daniels-Holg 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of Ricci flow with scaling invariant estimates

    Man-Chun Lee

    Topics Boundaries & non-compact, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “unbounded curvature”, “uniqueness theorem”, “Chen”

  2. Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds

    Dasong Li and John Man Shun MaWider flows

    Topics Boundaries & non-compact, Stability & uniqueness · Same author John Man-shun Ma · Shared terms “backwards uniqueness”, “uniqueness theorem”, “possibly”

  3. Note on the backwards uniqueness of mean curvature flow

    Zhuhong ZhangWider flows

    Topics Stability & uniqueness · Shared terms “backwards uniqueness”, “uniqueness theorem”

  4. On the uniqueness of Ricci flow

    Man-Chun Lee

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

  5. Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics

    Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “possibly unbounded”, “unbounded curvature”

  6. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Ricci flow · Same author Man-Chun Lee · Shared terms “possibly unbounded”, “unbounded curvature”

  7. Manifolds with small curvature concentration

    Pak-Yeung Chan, Shaochuang Huang and Man-Chun Lee

    Topics Ricci flow · Same author Man-Chun Lee · Shared terms “possibly unbounded”, “unbounded curvature”

  8. Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

    J. M. Daniels-Holgate and Or HershkovitsWider flows

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

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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 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.

  3. 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.