Paper · Nov 9, 2023

New advances on the existence and regularity of Brakke's mean curvature flows

Salvatore Stuvard

math.APWider flowsarXiv:2311.05370
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Abstract

In this survey paper, I discuss some recent progress on the existence and regularity of Brakke flows. These include: an "end-time version" of Brakke's local regularity theorem, which allows to extend the validity of the celebrated regularity theorem by White from limits of smooth mean curvature flows to arbitrary Brakke flows; a global-in-time existence theorem for multi-phase Brakke flows of grain boundaries satisfying suitable regularity in time, in both the unconstrained and the fixed boundary settings, with applications to Plateau's problem for the latter; and the proof that branching singularities of minimal surfaces are a trigger for dynamical instability, in the sense that they may be "perturbed away" by a non-trivial canonical Brakke flow. This note is an extended version of a talk given by the author at the MATRIX Research Institute on the occasion of the workshop entitled "Minimal surfaces and geometric flows: interaction between the local and the nonlocal worlds".

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Submitted
Nov 9, 2023
Subjects
math.AP, math.DG

Comments: Survey paper: extended version of a talk given at the MATRIX Research Institute (Creswick, Victoria, Australia) during the workshop on "Minimal surfaces and geometric flows: interaction between the local and the nonlocal worlds", Jan/Feb 2023. 14 pages. To appear on the MATRIX Annals Book Series

Mean curvature flowSingularitiesWeak & singular flowsLong-time existence & convergenceMinimal & CMC surfacesPhase fields & multiphase flows

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  1. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Brakke”

  2. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows

    Topics Minimal & CMC surfaces, Weak & singular flows, Singularities · Shared terms “Brakke flows”

  3. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Brakke”, “multi phase”

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

    Javier HervásWider flows

    Topics Weak & singular flows, Singularities · Shared terms “Brakke flows”, “local regularity”, “fixed boundary”

  5. Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn

    J. M. Daniels-HolgateWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Brakke”, “White”

  6. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Brakke”

  7. Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators

    Jingwen ChenWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Brakke flows”

  8. Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system

    Andrea PoiattiWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows, Long-time existence & convergence

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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

  2. 6A.13
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    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.