Minimal submanifolds as energy concentration sets: from phase transitions to gauge theoretic models

An important problem in calculus of variations and geometric analysis is the construction of minimal submanifolds, namely critical points for the area, due to their role in relating curvature and topology of the ambient. In the last three decades, a fruitful way to approximate the area functional in low codimension is to interpret submanifolds as the 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, as well as the new key difficulties. This talk is based on collaborations with Stern and Parise-Stern.