Singularities for harmonic maps

In this talk we introduce briefly the problem of regularity for harmonic maps and present the new regularity results proved for the singular sets of minimizing and stationary harmonic maps in collaboration with Aaron Naber (see arXiv:1504.02043).

We prove that the singular set of a minimizing harmonic map is rectifiable with effective n-3 volume estimates. The results are based on an improved quantitative stratification technique, which consists in a detailed analysis of the symmetries and almost symmetries of the map u and its blow-ups at different scales, and rely on a new $W^{1,p}$ version of Reifenberg's topological disk theorem. The application of this theorem in the situation of harmonic maps hinges on the monotonicity formula for the normalized energy.