ERC Starting Grant 101165368 - Magnetic "Minimal submanifolds in arbitrary geometries as nodal sets: towards higher codimension"

Principal Investigator: Alessandro Pigati

Abstract: 

The aim of this research project is to explore new variational ideas to produce and study minimal submanifolds in arbitrary curved ambients. These objects are surfaces or higher dimensional shapes which are stationary points for the area functional. While a lot is known by now for hypersurfaces I plan to investigate the situation in higher codimension using energies inspired from physics such as models of superconductivity as well as novel variational objects called parametrized varifolds. More in detail an energy which approximates well the area in codimension two is the abelian Higgs model which was first proposed in relation to this question in a joint work with D. Stern. The gradient flow of this energy is the natural approach to produce minimizers and critical points whose energy concentration set resembles a minimal submanifold: together with my collaborators we plan to refine its study adapting it to the Lagrangian setting and showing that it extends the classical mean curvature flow. We also plan to study regularity properties of stable critical points which are those arising from variational constructions. Moreover I will explore other candidate energies such as the non-abelian model in codimension three among others. I also plan to extend the regularity theory for another kind of object that I started investigating some years ago together with T. Rivière namely parametrized varifolds a weak notion of minimal surface retaining a parametrization. I will look at the Legendrian setting which appears naturally in certain special situations as well as the case of dimension higher than two.