Paper · Oct 12, 2021

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Reto Buzano, Huy The Nguyen, Mario B. Schulz

math.DGWider flowsarXiv:2110.06027v2
PDF Abstract page

As on arXiv

Abstract

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

Comments: Final version with minor modifications, to appear in Crelle

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfacesNumerics & visualization

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 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 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)6Cheng 2021 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)4Chu 2023 Self-shrinkers whose asymptotic cones fatten (Daniel Ketover, 2024)1Ketover 2024 Ends of (singular) Ricci shrinkers (Alessandro Bertellotti and Reto Buzano, 2024)3Bertellotti 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)7Chu 2024 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)8Xie 2025 The global structure of bi-rotational self-shrinkers (Gregório Silva Neto, 2026)2Neto 2026 The global structure of bi-rotational self-expanders (Gregŕio Silva Neto, 2026)5Neto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows PDF

    Topics Numerics & visualization, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “Ilmanen conjectured”, “conical ends”, “asymptotically conical”

  2. The global structure of bi-rotational self-shrinkers

    Gregório Silva NetoWider flows PDF

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

  3. Ends of (singular) Ricci shrinkers

    Alessandro Bertellotti and Reto Buzano PDF

    Topics Boundaries & non-compact · Same author Reto Buzano · Shared terms “conical ends”, “asymptotically conical”, “sequence”

  4. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Self-shrinkers & translators

  5. The global structure of bi-rotational self-expanders

    Gregŕio Silva NetoWider flows PDF

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

  6. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact

  7. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “min max”, “conjectured”

  8. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows PDF

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

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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 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.