Abstract:
We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern. The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. After starting with a general overview of the codimension one theory, we will move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied in gauge theory. We will explain to what extent the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interpret the former as a relaxation/regularization of the latter. We will mention some elements of the proof, with special emphasis on the role played by the gradient flow.