Codimension two area and Yang-Mills-Higgs
While the calculus of variations of the area of hypersurfaces is becoming more and more understood, with recent developments in the construction of unstable critical points, little is known for higher codimension. In this talk we present an approach to the codimension two, based on the Yang-Mills-Higgs energy for Hermitian line bundles, in the self-dual case. This functional, which is well known in gauge theory, turns out to be a good relaxation of the (codimension two) area functional, in a similar way as the Allen-Cahn functional in codimension one: in joint work with Daniel Stern we show that, if one uses scalings which preserve the self-duality, the energy of critical points concentrates along a minimal variety. We also discuss the corresponding Gamma-convergence theory, which is joint work with Daniel Stern and Davide Parise.