On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n geq 4$. For minimizing harmonic maps $uin W^{1,2}(Omega,mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Liebs linear law, namely [mathcal{H}^{n-3}(textrm{sing} u) le C int_{partial Omega} | abla_T u|^{n-1} ,dmathcal{H}^{n-1};] (2) An extension of Hardt and Lins stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.