Paper · Aug 9, 2021

Compactness and rigidity of self-shrinking surfaces

Tang-Kai Lee

math.DGWider flowsarXiv:2108.03919v3
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Abstract

The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.

Comments: 14 pages; some typos corrected

Mean curvature flowSelf-shrinkers & translatorsLimit spaces & RCDStability & uniqueness

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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 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 Rigidity results for mean curvature flow graphical translators moving in non-graphical direction (John Man Shun Ma, Yuan Shyong Ooi and Juncheol Pyo, 2022)6Ma 2022 Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2022)7Bourni 2022 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)3Martín 2023 Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow (Debora Impera and Michele Rimoldi, 2023)8Impera 2023 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)4Daniels-Holg 2025 Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers (Peng Peng, 2026)5Peng 2026 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)2Li 2026 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)1Braxton 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A stable self-shrinking Möbius bundle in

    Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Tang-Kai Lee · Shared terms “self shrinking”, “higher codimension”

  2. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

    Fagui Li and Yuhang ZhaoWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self shrinking”, “shrinking surfaces”

  3. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

    Topics Limit spaces & RCD, Self-shrinkers & translators, Stability & uniqueness · Shared terms “compactness”

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

    J. M. Daniels-Holgate and Or HershkovitsWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self shrinking”, “low entropy”

  5. Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

    Peng PengWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self shrinking”

  6. Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

    John Man Shun Ma, Yuan Shyong Ooi and Juncheol PyoWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness

  7. Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows PDF

    Topics Limit spaces & RCD · Shared terms “compactness”

  8. Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow

    Debora Impera and Michele RimoldiWider flows PDF

    Topics Self-shrinkers & translators, Stability & 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.

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.