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

TL;DR

This paper extends key results on the size and stability of singular sets of minimizing harmonic maps into the 2-sphere to higher dimensions, providing bounds and stability criteria.

Contribution

It generalizes Almgren and Lieb's linear law and Hardt and Lin's stability theorem to dimensions four and higher for harmonic maps into the 2-sphere.

Findings

Bound on the Hausdorff measure of the singular set in higher dimensions.

Stability of the singular set size under boundary perturbations.

Extension of classical theorems to n-dimensional domains.

Abstract

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. 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 \[\mathcal{H}^{n-3}(\textrm{sing} u) \le C \int_{\partial \Omega} |\nabla_T u|^{n-1} \,d\mathcal{H}^{n-1};\] (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.