Quantitative estimates for the singular strata of minimizing Harmonic maps

Regularity properties for minimizing harmonic maps between Riemannian manifolds have been known since the classical work of Schoen and Uhlenbeck (1982); in that context, an estimate on the Hausdorff dimension of the singular set S(u) is given. In particular, it is shown that dim(S(u)) is atmost n-3, where n is the dimension of the domain manifold. Simple examples show that this inequality can actually be an equality. In my thesis work, developed at the University of Zürich under the supervision of Prof. Camillo de Lellis and Dr. Daniele Valtorta, I am looking deeper into some more recent quantitative results, which describe precisely the structure of the singular set: firstly, the work of Cheeger and Naber (2013) permits to obtain an estimate of the Minkowski dimension of S(u); secondly, we follow the techniques of Naber and Valtorta (2017) to gain an upper bound on the Minkowski content and some information on the rectifiability of S(u).