Regularity of harmonic maps into Teichmüller space

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