Singular structures in geometric variational problems

Variational problems are concerned with determining the state of minimal energy (broadly construed) of a given system within a competition class of admissible states the system can achieve. It is known that solutions to variational problems arising in Geometry and Physics may, in general, exhibit singularities. A fine analysis of the size and structure of such singular sets is of pivotal importance, both from the purely theoretical perspective, and in view of the applications, in particular as a confirmation of the suitability of the variational model towards a correct description of the observed phenomena.

In this talk, I will describe my work on a variety of aspects concerning the physical relevance, the analytic properties, and the evolution of the singular structures arising in the solutions to some geometric variational models pertaining to the description of physical systems governed by surface tension-type energies, with an emphasis on (measure-theoretic generalizations of) minimal surfaces and mean curvature flows.