Paper · Jul 1, 2024

Self-shrinkers whose asymptotic cones fatten

Daniel Ketover

math.DGWider flowsarXiv:2407.01240v2
PDF Abstract page

As on arXiv

Abstract

For each positive integer we use variational methods to construct a genus self-shrinker in with entropy less than and prismatic symmetry group . For sufficiently large, the self-shrinker has two graphical asymptotically conical ends and the sequence converges on compact subsets to a plane with multiplicity two as . Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995 based on numerical experiments. Using these surfaces as initial conditions for large , we obtain examples of mean curvature flows in with smooth initial non-compact data that evolve non-uniquely after their first singular time.

Comments: Added additional references to recent related work

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactNumerics & 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)7Cheng 2021 Noncompact self-shrinkers for mean curvature flow with arbitrary genus (Reto Buzano, Huy The Nguyen and Mario B. Schulz, 2021)1Buzano 2021 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)6Garfinkle 2022 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)3Riedler 2022 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)2Chu 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)8Chodosh 2023 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)4Xie 2025 The global structure of bi-rotational self-shrinkers (Gregório 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. Noncompact self-shrinkers for mean curvature flow with arbitrary genus

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

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

  2. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “initial condition”

  3. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “Angenent”, “multiplicity”

  4. 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”

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

    Gregório Silva NetoWider flows PDF

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

  6. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows PDF

    Topics Numerics & visualization, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “numerical”, “sufficiently”

  7. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

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

  8. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows PDF

    Topics Boundaries & non-compact · Shared terms “cone fattens”, “asymptotically conical”, “Ilmanen”

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) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

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.