Paper · Apr 8, 2020

Bounded Diameter Under Mean Curvature Flow

Wenkui Du

math.DGWider flowsarXiv:2004.03769
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Abstract

We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, assuming Ilmanen's multiplicity one conjecture and no cylinder conjecture, we conclude that in the two-dimensional case, the diameter always stays bounded. We also obtain sharp bound for the curvature. The key ingredients for our proof are the Lojasiewicz inequalities by Colding-Minicozzi and Chodosh-Schulze, and the solution of the mean-convex neighbourhood conjecture by Choi, Haslhofer, Hershkovits and White. Our results improve the prior results by Gianniotis-Haslhofer, where diameter and curvature control has been obtained under the more restrictive assumption that the flow is globally two-convex.

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Submitted
Apr 8, 2020
Subjects
math.DG
Mean curvature flowSingularitiesSurfaces

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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 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 Diameter and curvature control under mean curvature flow (Panagiotis Gianniotis and Robert Haslhofer, 2017)2Gianniotis 2017 On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature (Haozhao Li and Bing Wang, 2018)8Li 2018 The blowdown of ancient noncollapsed mean curvature flows (Wenkui Du and Robert Haslhofer, 2021)4Du 2021 Hearing the shape of ancient noncollapsed flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2021)6Du 2021 A nonexistence result for rotating mean curvature flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2022)5Du 2022 Rigidity of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Jingze Zhu, 2025)7Choi 2025 The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow (Yiqi Huang and Wenshuai Jiang, 2025)3Huang 2025 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)1Bang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

    Kyeongho Bang, Eric Chen and Wenkui Du

    Topics Surfaces, Singularities · Same author Wenkui Du · Shared terms “bounded diameter”, “uniformly bounded”

  2. Diameter and curvature control under mean curvature flow

    Panagiotis Gianniotis and Robert HaslhoferWider flows

    Topics Mean curvature flow · Shared terms “curvature control”, “diameter”, “stays uniformly”

  3. The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

    Yiqi Huang and Wenshuai JiangWider flows

    Topics Singularities · Shared terms “bounded diameter”, “intrinsic diameter”, “uniformly bounded”

  4. The blowdown of ancient noncollapsed mean curvature flows

    Wenkui Du and Robert HaslhoferWider flows

    Topics Singularities · Same author Wenkui Du · Shared terms “Haslhofer”, “Choi”, “Hershkovits”

  5. A nonexistence result for rotating mean curvature flows in

    Wenkui Du and Robert HaslhoferWider flows

    Topics Singularities · Same author Wenkui Du · Shared terms “Colding Minicozzi”

  6. Hearing the shape of ancient noncollapsed flows in

    Wenkui Du and Robert HaslhoferWider flows

    Topics Singularities · Same author Wenkui Du · Shared terms “Hershkovits”, “Choi”, “White”

  7. Rigidity of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Jingze ZhuWider flows

    Topics Singularities · Same author Wenkui Du · Shared terms “Haslhofer”, “Choi”

  8. On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature

    Haozhao Li and Bing WangWider flows

    Topics Singularities · Shared terms “Ilmanen multiplicity”, “singular time”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.