ncts

The study of singularities of minimal surfaces is a fundamental problem in Geometric Analysis, which, for decades, has been fuelling some beautiful research in Differential Geometry, Geometric Measure Theory, Calculus of Variations, and PDE. It is widely known that a class of singularities which is particularly hard to treat is that of “branched” or “flat” singularities: these cannot be detected at the classical blow-up / linearization level, because their tangent cones are smooth planes with multiplicity larger than one. Yet, flat singularities are unavoidable even in case of hypersurfaces, and even if stability is assumed. In this talk, I will prove, for minimal hypersurfaces, that if blow-ups at a flat singularity converge to the tangent plane at a suitable (mild) rate then the singularity is dynamically unstable, in the sense that it can be perturbed away with a non-trivial, area reducing mean curvature flow (in the sense of Brakke). The rate of convergence we need to assume is satisfied in all known examples, and it can be proved to hold for large classes of solutions. I will then argue that the mean curvature flow may be used as a selection principle for “well-behaved” solutions to Plateau’s problem, even possibly in the case of very wild boundaries. This is joint work with Yoshihiro Tonegawa (Institute of Science Tokyo).