Calculus of Variations and Geometric Measure Theory

J. Hirsch - S. Stuvard - D. Valtorta

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

created by stuvard on 08 Aug 2017
modified on 27 Jun 2019


Published Paper

Inserted: 8 aug 2017
Last Updated: 27 jun 2019

Journal: Trans. Amer. Math. Soc.
Volume: 371
Number: 6
Pages: 4303-4352
Year: 2019
Doi: 10.1090/tran/7595

ArXiv: 1708.02116 PDF
Links: Webpage


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.