Rectifiability of the singular set of multiple valued energy minimizing harmonic maps

TL;DR

This paper proves that the singular set of Dirichlet-minimizing Q-valued harmonic maps into smooth manifolds is always (m-3)-rectifiable with uniform bounds, and explores how target curvature affects map continuity.

Contribution

It extends rectifiability results to multiple valued harmonic maps and investigates the impact of target curvature on regularity.

Findings

Singular set is (m-3)-rectifiable with Minkowski bounds.

Non-positive curvature of the target does not guarantee map continuity.

Results generalize known properties from single valued to multiple valued harmonic maps.

Abstract

In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single valued case, we prove that the target $\mathcal{N}$ being non-positively curved but not simply connected does not imply continuity of the map.