Paper · Mar 1, 2018

Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

Or Hershkovits, Brian White

math.DGWider flowsarXiv:1803.00637v2
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Abstract

Let be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial homology. We show that the entropy of is greater than or equal to the entropy of a round -sphere, and that if equality holds, then is a round -sphere in .

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Submitted
Mar 1, 2018
Last revised
Oct 14, 2018 (v2)
Subjects
math.DG
Journal
Geom. Topol. 23 (2019) 1611-1619
DOI
10.2140/gt.2019.23.1611

Comments: 7 pages. The new version (Oct 13, 2018) incorporates a few changes and clarifications suggested by the Geometry and Topology referees

Mean curvature flowSelf-shrinkers & translators

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 entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)4Zhu 2016 Non-fattening of mean curvature flow at singularities of mean convex type (Or Hershkovits and Brian White, 2017)3Hershkovits 2017 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)6Baldauf 2018 Ancient low entropy flows, mean convex neighborhoods, and uniqueness (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2018)7Choi 2018 Ancient asymptotically cylindrical flows and applications (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al., 2019)1Choi 2019 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)2White 2022 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)8Zhao 2025 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)5Daniels-Holg 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient asymptotically cylindrical flows and applications

    Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al.Wider flows

    Topics Self-shrinkers & translators · Same authors Or Hershkovits, Brian White

  2. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows

    Topics Self-shrinkers & translators · Same author Brian White

  3. Non-fattening of mean curvature flow at singularities of mean convex type

    Or Hershkovits and Brian WhiteWider flows

    Topics Mean curvature flow · Same authors Or Hershkovits, Brian White

  4. On the entropy of closed hypersurfaces and singular self-shrinkers

    Jonathan J. ZhuWider flows

    Topics Self-shrinkers & translators

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

    J. M. Daniels-Holgate and Or HershkovitsWider flows

    Topics Self-shrinkers & translators · Same author Or Hershkovits

  6. Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

    Julius Baldauf and Ao SunWider flows

    Topics Mean curvature flow · Shared terms “sharp entropy”, “entropy bound”

  7. Ancient low entropy flows, mean convex neighborhoods, and uniqueness

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Self-shrinkers & translators · Same author Or Hershkovits

  8. A gap Theorem on closed self-shrinkers of mean curvature flow

    Yuhang ZhaoWider flows

    Topics Self-shrinkers & translators

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) 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)

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.