Paper · Sep 4, 2017

Note on the backwards uniqueness of mean curvature flow

Zhuhong Zhang

math.DGWider flowsarXiv:1709.00798
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Abstract

In this note, we will show a backwards uniqueness theorem of the mean curvature flow with bounded second fundamental form in arbitrary codimension.

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Submitted
Sep 4, 2017
Subjects
math.DG
Journal
SCIENCE CHINA Mathematics, 2018
DOI
10.1007/s11425-017-9231-4

Comments: 8 pages

Mean curvature flowStability & uniqueness

Its neighbourhood

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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 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 Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures (Man-Chun Lee and John Man-shun Ma, 2017)3Lee 2017 Parabolic frequency on Ricci flows (Julius Baldauf and Dain Kim, 2022)5Baldauf 2022 Parabolic frequency for the mean curvature flow (Julius Baldauf and Tang-Kai Lee, 2022)4Baldauf 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)7Lee 2023 Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds (Dasong Li and John Man Shun Ma, 2024)2Li 2024 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)6Chen 2024 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)1Daniels-Holg 2025 Well-posedness of mean curvature flow (Yongheng Han, 2026)8Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

    J. M. Daniels-Holgate and Or HershkovitsWider flows

    Topics Stability & uniqueness · Shared terms “backwards uniqueness”

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

    Dasong Li and John Man Shun MaWider flows

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

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

    Man-Chun Lee and John Man-shun Ma

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

  4. Parabolic frequency for the mean curvature flow

    Julius Baldauf and Tang-Kai LeeWider flows

    Topics Stability & uniqueness · Shared terms “backwards uniqueness”

  5. Parabolic frequency on Ricci flows

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    Topics Stability & uniqueness · Shared terms “backwards uniqueness”

  6. A rigidity theorem of ancient solutions to the mean curvature flow in codimension one

    Qun Chen and Hongbing QiuWider flows

    Topics Stability & uniqueness · Shared terms “fundamental form”

  7. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows

    Topics Stability & uniqueness

  8. Well-posedness of mean curvature flow

    Yongheng HanWider flows

    Topics Stability & uniqueness · Shared terms “fundamental form”

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.

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)

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.