Improved regularity for nodal sets of Abelian Yang-Mills-Higgs equations.

We consider Yang-Mills-Higgs equations with U(1) gauge group. There is a deep relation between the adiabatic limit of a sequence of this physical PDEs and geometric PDE of minimal submanifolds. It is known that the energy measures are converging to a codimension 2 stationary varifold and the energy functional is converging to the codimension 2 mass. When the ambient dimension is \leq 4 or the sequence is minimizing, we can improve the weak convergence above and obtain strong regularity for the nodal sets that they are converging to the limit minimal submanifold with uniform $C^{2,\alpha}$ bounds. This is joint work with Huy Nguyen.