Paper · Aug 26, 2021

Quantitative long range curvature estimate for mean curvature flow

Jingze Zhu

math.DGWider flowsarXiv:2108.11852
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Abstract

We prove that smooth convex -noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate for any pair of . In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.

Mean curvature flowAncient solutions(Non)collapsing

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 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 A nonexistence result for rotating mean curvature flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2022)4Du 2022 Spectral quantization for ancient asymptotically cylindrical flows (Wenkui Du and Jingze Zhu, 2022)2Du 2022 Enhanced profile estimates for ovals and translators (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2023)7Choi 2023 Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image (Hongbing Qiu and Y. L. Xin, 2023)5Qiu 2023 Rigidity of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Jingze Zhu, 2025)1Choi 2025 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)3Lee 2025 Classification of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Ziyi Zhao, 2026)8Choi 2026 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)6Deng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Jingze ZhuWider flows PDF

    Topics (Non)collapsing, Ancient solutions · Same author Jingze Zhu · Shared terms “noncollapsed”

  2. Spectral quantization for ancient asymptotically cylindrical flows

    Wenkui Du and Jingze ZhuWider flows PDF

    Topics (Non)collapsing, Ancient solutions · Same author Jingze Zhu · Shared terms “noncollapsed”

  3. Planarity and convexity for pinched ancient solutions of mean curvature flow

    Tang-Kai Lee, Keaton Naff and Jingze Zhu PDF

    Topics Ancient solutions · Same author Jingze Zhu

  4. A nonexistence result for rotating mean curvature flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics (Non)collapsing, Ancient solutions · Shared terms “noncollapsed”

  5. Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

    Hongbing Qiu and Y. L. XinWider flows PDF

    Topics Ancient solutions · Shared terms “curvature estimates”

  6. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics (Non)collapsing, Ancient solutions · Shared terms “noncollapsed ancient”

  7. Enhanced profile estimates for ovals and translators

    Kyeongsu Choi, Robert Haslhofer and Or Hershkovits PDF

    Topics (Non)collapsing, Ancient solutions · Shared terms “noncollapsed”

  8. Classification of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Ziyi ZhaoWider flows PDF

    Topics (Non)collapsing, Ancient solutions · Shared terms “noncollapsed”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) (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)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

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

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.