Proving existence of minimal submanifolds (i.e. critical points of the area functional) in various settings has been one of the driving forces of the development of modern calculus of variations, and geometric measure theory, with remarkable applications in differential geometry. In the 1960s Almgren developed a far-reaching technique to prove existence of (measure theoretic versions of) these objects. While effective, Almgren's technique was notoriously complex which prompted the emergence in recent years of an alternative PDE-based method. The underlying idea is to construct minimal submanifolds as limits of nodal sets of critical points of functionals arising from the gradient theory of phase transitions and the theory of superconductors. After starting with a general overview, I will explain how the work of my collaborators and I fits into the broader picture. In particular, we will start with the Allen-Cahn functional and the codimension one theory discussing the construction of free boundary minimal surfaces, i.e. minimal submanifolds meeting the boundary orthogonally. We will then move to higher codimensions (specifically codimensions 2 and 3), where the theory is much less developed. We will introduce the Yang-Mills-Higgs functionals, in both the abelian and non-abelian settings. In particular, we will focus on what happens to their gradient flows, and their variational properties. I will highlight other successes of this theory, and point to some open problems along the way. The content of this talk is based on joint works with Martin Li, Lorenzo Sarnataro, Alessandro Pigati, and Daniel Stern.
