Paper · May 26, 2018

Translators asymptotic to cylinders

Or Hershkovits

math.DGarXiv:1805.10553
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Abstract

We show that the Bowl soliton in is the unique translating solutions of the mean curvature flow which has the family of shrinking cylinders as an asymptotic shrinker at . As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soliton. The crucial point is that we do not make any global convexity assumption, while as the same time, the asymptotic requirement is very weak.

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Submitted
May 26, 2018
Subjects
math.DG, math.AP
Mean curvature flowRicci solitonsSelf-shrinkers & translators

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 Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow (Or Hershkovits and Brian White, 2018)8Hershkovits 2018 Ancient low entropy flows, mean convex neighborhoods, and uniqueness (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2018)2Choi 2018 Ancient asymptotically cylindrical flows and applications (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al., 2019)5Choi 2019 Classification of noncollapsed translators in ℝ⁴ (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2021)1Choi 2021 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)4Sun 2023 Enhanced profile estimates for ovals and translators (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2023)3Choi 2023 Bowl Soliton Asymptotics and Applications (Sathya Rengaswami and José Torres Santaella, 2023)7Rengaswami 2023 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)6Daniels-Holg 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of noncollapsed translators in

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Self-shrinkers & translators · Same author Or Hershkovits

  2. 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 · Shared terms “bowl soliton”, “limit flows”, “shrinking cylinder”

  3. Enhanced profile estimates for ovals and translators

    Kyeongsu Choi, Robert Haslhofer and Or Hershkovits

    Topics Self-shrinkers & translators · Same author Or Hershkovits

  4. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows

    Topics Self-shrinkers & translators · Shared terms “translator asymptotic”

  5. Ancient asymptotically cylindrical flows and applications

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

    Topics Self-shrinkers & translators · Same author Or Hershkovits · Shared terms “bowl soliton”, “shrinking cylinder”

  6. 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

  7. Bowl Soliton Asymptotics and Applications

    Sathya Rengaswami and José Torres SantaellaWider flows

    Topics Self-shrinkers & translators · Shared terms “bowl soliton”

  8. Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

    Or Hershkovits and Brian WhiteWider flows

    Topics Self-shrinkers & translators · Same author Or Hershkovits

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 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.