Paper · May 4, 2020

An unknottedness result for noncompact self shrinkers

Alexander Mramor

math.DGarXiv:2005.01688v11
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Abstract

In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. The mean curvature flow itself is used in the argument presented.

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Submitted
May 4, 2020
Last revised
Mar 11, 2024 (v11)
Subjects
math.DG, math.GT

Comments: Further revised version, to appear in Crelle's

Mean curvature flowRicci solitonsSelf-shrinkers & translatorsBoundaries & non-compact

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 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)8Mramor 2018 Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical End (Siao-Hao Guo, 2019)7Guo 2019 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)4Khan 2020 An unknottedness result for self shrinkers with multiple ends (Alexander Mramor, 2020)1Mramor 2020 Noncompact self-shrinkers for mean curvature flow with arbitrary genus (Reto Buzano, Huy The Nguyen and Mario B. Schulz, 2021)2Buzano 2021 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)5Lee 2024 Topology of gradient Ricci shrinkers via weighted L² cohomology (Fei He, 2026)6He 2026 On knotted self expanders of the mean curvature flow in R³ (Alexander Mramor and Shengwen Wang, 2026)3Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An unknottedness result for self shrinkers with multiple ends

    Alexander MramorWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same author Alexander Mramor · Shared terms “unknottedness”, “conical ends”, “asymptotically conical”

  2. Noncompact self-shrinkers for mean curvature flow with arbitrary genus

    Reto Buzano, Huy The Nguyen and Mario B. SchulzWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “noncompact self”, “conical ends”, “conjecturally”

  3. On knotted self expanders of the mean curvature flow in

    Alexander Mramor and Shengwen Wang

    Topics Self-shrinkers & translators, Ricci solitons · Same author Alexander Mramor

  4. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “asymptotically conical”

  5. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao

    Topics Self-shrinkers & translators, Boundaries & non-compact, Ricci solitons · Shared terms “asymptotically conical”

  6. Topology of gradient Ricci shrinkers via weighted cohomology

    Fei He

    Topics Self-shrinkers & translators, Ricci solitons

  7. Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical End

    Siao-Hao Guo

    Topics Self-shrinkers & translators, Ricci solitons · Shared terms “conical ends”

  8. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider flows

    Topics Self-shrinkers & translators · Same author Alexander Mramor

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.