Paper · Oct 27, 2017

Regularity and stability results for the level set flow via the mean curvature flow with surgery

Alexander Mramor

math.DGWider flowsarXiv:1710.09989v4
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Abstract

In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.

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Submitted
Oct 27, 2017
Last revised
Apr 24, 2019 (v4)
Subjects
math.DG, math.AP

Comments: Revisions incorporating referee's comments made. To appear in CAG

Mean curvature flowSurgeryWeak & singular flowsStability & uniqueness

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 A level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)3Giga 2016 A finiteness theorem via the mean curvature flow with surgery (Alexander Mramor, 2017)2Mramor 2017 Level set flow in 3D steady gradient Ricci solitons (Chih-Wei Chen and Kuo-Wei Lee, 2017)4Chen 2017 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)1Mramor 2018 High Codimension Mean Curvature Flow with Surgery (Stephen Lynch and Huy The Nguyen, 2020)8Lynch 2020 Mean curvature flow in homology and foliations of hyperbolic 3-manifolds (Marco A. M. Guaraco, Vanderson Lima and Franco Vargas Pallete, 2021)7Guaraco 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)6Sun 2022 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)5Chodosh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider flows

    Topics Surgery · Same author Alexander Mramor

  2. A finiteness theorem via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Surgery · Same author Alexander Mramor

  3. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “level set”, “set flow”

  4. Level set flow in 3D steady gradient Ricci solitons

    Chih-Wei Chen and Kuo-Wei Lee

    Topics Weak & singular flows · Shared terms “set flow”, “level set”

  5. Mean Curvature Flow from Conical Singularities

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

    Topics Weak & singular flows · Shared terms “set flow”, “level set”

  6. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue

    Topics Weak & singular flows · Shared terms “set flow”, “level set”

  7. Mean curvature flow in homology and foliations of hyperbolic -manifolds

    Marco A. M. Guaraco, Vanderson Lima and Franco Vargas PalleteWider flows

    Topics Surgery, Weak & singular flows · Shared terms “set flow”, “level set”

  8. High Codimension Mean Curvature Flow with Surgery

    Stephen Lynch and Huy The NguyenWider flows

    Topics Surgery, Weak & singular flows

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  3. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.