Paper · Feb 11, 2016

The free-boundary Brakke flow

Nick Edelen

math.DGWider flowsarXiv:1602.03614v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White to the free-boundary setting. We use Ilmanen's elliptic regularization procedure to prove existence of free-boundary Brakke flows.

View PDF View HTML arXiv page
Submitted
Feb 11, 2016
Last revised
Nov 16, 2017 (v3)
Subjects
math.DG

Comments: fixed some details, references; accepted to Crelle

Mean curvature flowWeak & singular flowsLimit spaces & RCDFree boundary & capillary

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 Regularity of minimal submanifolds and mean curvature flows with a common free boundary (Brian Krummel, 2016)8Krummel 2016 Equality of the usual definitions of Brakke flow (Ananda Lahiri, 2017)6Lahiri 2017 Mean convex mean curvature flow with free boundary (Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki et al., 2019)1Edelen 2019 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)7Bourni 2022 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)5Tashiro 2024 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)3Bao 2026 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)4Hervás 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)2Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean convex mean curvature flow with free boundary

    Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki et al.Wider flows

    Topics Free boundary & capillary, Limit spaces & RCD · Same author Nick Edelen · Shared terms “boundary setting”, “White”

  2. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “boundary Brakke”, “Brakke flows”

  3. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “boundary Brakke”, “Brakke flows”, “boundary setting”

  4. Spacelike Brakke flows with boundary in pseudo-Euclidean space

    Javier HervásWider flows

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “Brakke flows”, “regularization procedure”, “Ilmanen elliptic”

  5. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “elliptic regularization”, “Brakke”, “Ilmanen”

  6. Equality of the usual definitions of Brakke flow

    Ananda LahiriWider flows

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “Brakke flows”

  7. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows

    Topics Free boundary & capillary

  8. Regularity of minimal submanifolds and mean curvature flows with a common free boundary

    Brian KrummelWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “Brakke 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.

Foundational paper in the Free boundary & capillary guide: Edelen: the free-boundary Brakke flow.

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.