On the size of the singular set of minimizing harmonic maps

TL;DR

This paper extends key results on the size and stability of singular sets in minimizing harmonic maps into manifolds, covering higher dimensions and weaker boundary regularity conditions.

Contribution

It generalizes Almgren and Lieb's linear law to dimensions n ≥ 4 and extends Hardt and Lin's stability theorem for maps into spheres, also considering weaker boundary regularity.

Findings

Bound on the Hausdorff measure of the singular set for n ≥ 4.

Stability of the singular set under boundary perturbations for maps into S^2.

Results hold under weaker boundary regularity assumptions in dimension 3.

Abstract

We consider minimizing harmonic maps $u$ from $\Omega \subset \mathbb{R}^n$ into a closed Riemannian manifold $\mathcal{N}$ and prove: (1) an extension to $n \geq 4$ of Almgren and Lieb's linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$ is finite, we have \[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial \Omega} |\nabla_T u|^{n-1} \,d \mathcal{H}^{n-1}; \] (2) an extension of Hardt and Lin's stability theorem. Namely, assuming that the target manifold is $\mathcal{N}=\mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s \in (\frac{1}{2},1]$ and $p \in [2,\infty)$ satisfying $sp \geq 2$. We also discuss sharpness.