Totally Geodesic Submanifolds of Teichmüller Space in Complex Dimension at Least Three

Let $N\subset {\mathcal{T}}_{g,n}$ be a connected complex totally geodesic submanifold. We prove that if ${\dim }_{\mathbb{C}}N\ge 3$, then $N$ is the image of an entire Teichmüller space under a covering construction induced by a finite branched cover of marked surfaces. This answers a question of Arana-Herrera and Wright.