Hodge metric completion of the Teichm\"uller space of Calabi-Yau manifolds

We prove that the Hodge metric completion of the Teichm\"uller space of polarized and marked Calabi-Yau manifolds is a complex affine manifold. We also show that the extended period map from the completion space is injective into the period domain, and that the completion space is a domain of holomorphy and admits a complete K\"ahler-Einstein metric.