Minimal submanifolds: parametrized surfaces, phase transitions, and gauge theoretic models

Seminar in Mathematics

Abstract: An important and popular problem in calculus of variations and geometric analysis, and in fact one which motivated the development of a large part of them, is the construction of minimal submanifolds. These are critical points for the area functional and, besides an intrinsic interest, their relevance is due to the fact that they sometimes allow to relate curvature and topology of the ambient.

While I will not attempt to survey all the different frameworks that have been developed so far, I will rather focus on my contributions to this topic. In the case of surfaces, a classical approach for disks and spheres is to exhibit and study critical parametrizations. In this setting, I will briefly discuss a series of results in collaboration with Rivière, dealing with a new weak notion of critical point which allows to have an arbitrary prescribed genus, as well as a satisfactory regularity theory, even under a Lagrangian or Legendrian constraint.

In the last few decades, another fruitful way to approximate the area functional in low codimension has been to interpret submanifolds as nodal sets of maps (or sections of vector bundles), critical for suitable physical energies or well-known lagrangians from gauge theory. Inspired by the well understood Allen-Cahn energy in codimension one, we will survey the situation in codimension two, where the abelian U(1) Higgs model has provided a successful framework (leaving the regularity theory aside), as well as the non-abelian SU(2) Yang-Mills-Higgs as a natural candidate in codimension three, emphasizing analogies and differences. This second part is based on collaborations with Stern and Parise-Stern.