Paper · Mar 31, 2020

Mean curvature flow with generic initial data

Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze

math.DGarXiv:2003.14344v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We show that the mean curvature flow of generic closed surfaces in avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

View PDF View HTML arXiv page
Submitted
Mar 31, 2020
Last revised
Apr 2, 2024 (v2)
Subjects
math.DG, math.AP

Comments: Final version, to appear in Invent. Math

Mean curvature flowRicci solitonsAncient solutionsSingularitiesStability & uniquenessBoundaries & non-compactLong-time existence & convergence

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)8Choi 2018 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)3Chodosh 2019 Ancient gradient flows of elliptic functionals and Morse index (Kyeongsu Choi and Christos Mantoulidis, 2019)7Choi 2019 Mean curvature flow with generic low-entropy initial data (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2021)4Chodosh 2021 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)1Chodosh 2023 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)2Chodosh 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)6Chodosh 2023 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)5Chodosh 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

    Topics Singularities · Same authors Otis Chodosh, Kyeongsu Choi, Felix Schulze · Shared terms “generic initial”, “generic closed”, “initial data”

  2. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows

    Topics Singularities · Same authors Otis Chodosh, Christos Mantoulidis, Felix Schulze · Shared terms “generic closed”, “low entropy”, “initial data”

  3. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same authors Otis Chodosh, Felix Schulze · Shared terms “asymptotically conical”

  4. Mean curvature flow with generic low-entropy initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows

    Topics Singularities · Same authors Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis · Shared terms “low entropy”, “initial data”, “closed surfaces”

  5. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows

    Topics Ancient solutions, Singularities · Same authors Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis

  6. Mean Curvature Flow from Conical Singularities

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

    Topics Boundaries & non-compact, Singularities · Same authors Otis Chodosh, Felix Schulze · Shared terms “asymptotically conical”

  7. Ancient gradient flows of elliptic functionals and Morse index

    Kyeongsu Choi and Christos MantoulidisWider flows

    Topics Ancient solutions · Same authors Kyeongsu Choi, Christos Mantoulidis

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

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Ancient solutions, Singularities, Stability & uniqueness · Same author Kyeongsu Choi · Shared terms “low entropy”, “spherical”

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.

Foundational paper in the Mean curvature flow guide: Chodosh–Choi–Mantoulidis–Schulze: generic closed surfaces avoid conical and non-spherical compact singularities.

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

    Comparing curvature across space and time.