The regularity of conformal target harmonic maps

We establish that any weakly conformal \(W^{1,2}\) map from a Riemann surface S into a closed oriented sub-manifold \(N^n\) of an euclidian space \({\mathbb {R}}^m\) realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form S into \(N^n\).