Codimension two min-max minimal submanifolds from PDEs

A fruitful method to construct min-max minimal hypersurfaces, in an ambient closed Riemannian manifold, is given by the Allen-Cahn functional and its natural rescalings. This approach produces minimal hypersurfaces as limiting interfaces in a phase transition between two different states. A natural attempt to carry out this in codimension two is to use the simplified Ginzburg-Landau functional (with no magnetic potential and no external field) and hope (roughly speaking) to capture a minimal submanifold as a limit of the vortices, as the repulsion parameter tends to infinity. It is known since two decades that one always gets a stationary rectifiable varifold in the limit, but it is an open problem to establish its integrality. After briefly describing the difficulties arising with this functional, we will see how they disappear looking instead at the Yang-Mills-Higgs action (or the full Ginzburg-Landau free energy), with the natural rescalings preserving self-duality, first proposed by Hong-Jost-Struwe in the two-dimensional setting. We will discuss the features of this relaxation of the codimension-two area, which become strikingly similar to those of Allen-Cahn: in particular, integrality of the limit varifold and exponential decay of the energy density (away from vortices) do hold. We also describe how to construct a (codimension two) min-max integer stationary varifold in any ambient closed manifold using this approach. This is joint work with Daniel Stern (Princeton University - University of Toronto).