The epsilon-regularity theorem for Brakke flows near triple junctions

Abstract: In a pioneering paper published on JDG in 1993, Leon Simon established a powerful method to demonstrate, among other things, the validity of the following result: if a multiplicity one minimal k-dimensional surface (stationary varifold) is sufficiently close, in the unit ball and in a weak measure-theoretic sense, to the stationary cone given by the union of three k-dimensional half-planes meeting along a (k-1)-dimensional subspace and forming angles of 120 degrees with one another, then, in a smaller ball, the surface must be a $C^{1,\alpha}$ deformation of the cone. In this talk, I will present the proof of a parabolic counterpart of this result, which applies to general classes of (possibly forced) weak mean curvature flows (Brakke flows). I will particularly focus on the need of an assumption, which is absent in the elliptic case, and which, on the other hand, is satisfied by both Brakke flows with multi-phase grain boundaries structure and by Brakke flows that are flows of currents mod 3: these are the main classes of Brakke flows for which a satisfactory existence theory is currently available and triple junction singularities are expected. In these cases, the theorem holds true unconditionally, and it implies uniqueness of multiplicity-one, backward-static triple junctions as tangent flows as well as a structure theorem on the singular set under suitable Gaussian density restrictions. This is a joint work with Yoshihiro Tonegawa (Institute of Science Tokyo).

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

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