Interior regularity for two-dimensional stationary \(Q\)-valued maps (Q6590642)

In this very interesting paper under review, the authors study the regularity theory of 2-dimensional multi-valued (``\(Q\)-valued'', following Almgren) stationary (harmonic) maps; i.e., the maps stationary with respect to both the outer and inner variations of the Dirichlet energy. More precisely, it is proved that a 2D \(Q\)-valued stationary map is Hölder continuous, and that the dimension of the singular set is at most 1 (which is optimal).\N\NIt was previously established in [\textit{C.-C. Lin}, J. Geom. Anal. 24, No. 3, 1547--1582 (2014; Zbl 1302.49050)] that a 2D \(Q\)-valued stationary map is continuous, by way of using Almgren's embedding. This paper simplifies and provides an intrinsic version of the arguments therein, and improves the result to Hölder continuity. In contrast to the regularity theory of Dir-minimizing (harmonic) maps, comparison with suitable competitors is not available in the setting of stationary maps. Also, counterexamples show that continuity does not hold in general for \textit{weakly} stationary maps.\N\NOne key ingredient of the proof is a new strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, which holds in every dimension. The proof also makes use of conformal mappings in dimension two. It remains open if the regularity theory established in this paper carries over to dimensions \(d \geq 3\).