Regularity of harmonic maps into Teichm\"uller space

We prove a regularity theorem for harmonic maps into Teichm\"uller space. More specifically, if $u$ is a harmonic map from a Riemannian domain to the metric completion of Teichm\"uller space with respect to the Weil-Petersson metric, and the image of $u$ intersects a stratum of the augmented Teichm\"uller space, then $u$ is entirely contained in this stratum. This extends Wolpert's result on the geodesic convexity of the augmented Teichm\"uller space to higher dimensions and generalizes the regularity result of Daskalopoulos and Mese by showing that the singular set of $u$ is empty.