Paper · Oct 5, 2019

A note on the selfsimilarity of limit flows

Beomjun Choi, Robert Haslhofer, Or Hershkovits

math.DGWider flowsarXiv:1910.02341
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Abstract

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity.

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Submitted
Oct 5, 2019
Subjects
math.DG, math.AP

Comments: 5 pages

Mean curvature flowAncient solutionsSingularities

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 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)3Choi 2019 A nonexistence result for wing-like mean curvature flows in ℝ⁴ (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2021)6Choi 2021 A note on blowup limits in 3d Ricci flow (Beomjun Choi and Robert Haslhofer, 2021)1Choi 2021 Classification of bubble-sheet ovals in ℝ⁴ (Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al., 2022)5Choi 2022 Enhanced profile estimates for ovals and translators (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2023)7Choi 2023 Classification of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Ziyi Zhao, 2026)8Choi 2026 Ancient cylindrical flows with fast or slow convergence (Kyeongsu Choi and Robert Haslhofer, 2026)4Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on blowup limits in 3d Ricci flow

    Beomjun Choi and Robert Haslhofer

    Topics Ancient solutions, Singularities · Same authors Beomjun Choi, Robert Haslhofer · Shared terms “blowup limits”, “spherical singularities”, “ovals occur”

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

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Ancient solutions, Singularities · Same authors Robert Haslhofer, Or Hershkovits · Shared terms “limit flows”, “convex neighborhood”, “mean convex”

  3. Ancient asymptotically cylindrical flows and applications

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

    Topics Ancient solutions, Singularities · Same authors Robert Haslhofer, Or Hershkovits · Shared terms “neck singularities”, “convex neighborhood”, “blowup limits”

  4. Ancient cylindrical flows with fast or slow convergence

    Kyeongsu Choi and Robert HaslhoferWider flows

    Topics Ancient solutions, Singularities · Same author Robert Haslhofer · Shared terms “neck singularities”, “neighborhood conjecture”, “convex neighborhood”

  5. Classification of bubble-sheet ovals in

    Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al.Wider flows

    Topics Ancient solutions · Same authors Beomjun Choi, Robert Haslhofer · Shared terms “ancient ovals”, “selfsimilar”

  6. A nonexistence result for wing-like mean curvature flows in

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Ancient solutions, Singularities · Same authors Robert Haslhofer, Or Hershkovits

  7. Enhanced profile estimates for ovals and translators

    Kyeongsu Choi, Robert Haslhofer and Or Hershkovits

    Topics Ancient solutions · Same authors Robert Haslhofer, Or Hershkovits · Shared terms “ancient ovals”

  8. Classification of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Ziyi ZhaoWider flows

    Topics Ancient solutions · Same author Beomjun Choi · Shared terms “ancient ovals”, “selfsimilar”, “neighborhood conjecture”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.