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

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 ${\rm BV}$ 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".