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

Abstract

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

33 pages, 1 figure