Paper · May 21, 2016

A local regularity theorem for mean curvature flow with triple edges

Felix Schulze, Brian White

math.DGWider flowsarXiv:1605.06592v2
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Abstract

Mean curvature flow of clusters of n-dimensional surfaces in R^n+k that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.

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Submitted
May 21, 2016
Last revised
Jun 6, 2017 (v2)
Subjects
math.DG, math.AP

Comments: Existence proof extended to cover existence of a Brakke flow, starting from a general cluster in arbitrary codimension. Added details in proof of initial smoothness. Presentation improved, references updated, some typos fixed. 25 pages

Mean curvature flowWeak & singular flowsHarmonic maps & DeTurck trick

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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 Non-fattening of mean curvature flow at singularities of mean convex type (Or Hershkovits and Brian White, 2017)2Hershkovits 2017 Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds (Felix Schulze, 2018)7Schulze 2018 On the regularity of Ricci flows coming out of metric spaces (Alix Deruelle, Felix Schulze and Miles Simon, 2019)1Deruelle 2019 Short time existence of the classical solution to the fractional Mean curvature flow (Vesa Julin and Domenico La Manna, 2019)8Julin 2019 Ancient asymptotically cylindrical flows and applications (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al., 2019)4Choi 2019 Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion (Helmut Abels, Felicitas Bürger and Harald Garcke, 2022)6Abels 2022 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)5Lotay 2022 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)3Chodosh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the regularity of Ricci flows coming out of metric spaces

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Harmonic maps & DeTurck trick, Weak & singular flows · Same author Felix Schulze

  2. Non-fattening of mean curvature flow at singularities of mean convex type

    Or Hershkovits and Brian WhiteWider flows

    Topics Weak & singular flows · Same author Brian White · Shared terms “flow starting”

  3. Mean Curvature Flow from Conical Singularities

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

    Topics Weak & singular flows · Same author Felix Schulze · Shared terms “short time”

  4. Ancient asymptotically cylindrical flows and applications

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

    Topics Weak & singular flows · Same author Brian White · Shared terms “Brakke flows”, “integral Brakke”

  5. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Weak & singular flows · Same author Felix Schulze · Shared terms “static union”

  6. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”

  7. Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

    Felix SchulzeWider flows

    Topics Weak & singular flows · Same author Felix Schulze

  8. Short time existence of the classical solution to the fractional Mean curvature flow

    Vesa Julin and Domenico La MannaWider flows

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”

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) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.