On the size of the singular set of minimizing harmonic maps

Minimizing harmonic maps (i.e. minimizers of the Dirichlet integral) with prescribed boudary conditions may have singularities. At the beginning of this talk, I will consider minimizing harmonic maps from 3-dimensional domains into the two dimensional sphere and present an extension of Almgren and Lieb’s linear law on the bound of the singular set as well as Hardt and Lin’s stability theorem for singularities. Next, I will discuss new higher dimensional counterparts of those theorems. This is joint work with Micha\l{} Mi\'{s}kiewicz and Armin Schikorra.