Classical results by Royden, Earle and Kra on the geometry of Teichmüller spaces of finite type surfaces imply that the isometry group for the Teichmüller metric, the biholomorphism group of Teichmüller space and the mapping class group of the underlying surface are all isomorphic. In other words, every isometry of Teichmüller space is induced by a homeomorphism of the underlying surface.
Since then, these results have been generalized in many directions. For example, by weakening isometries to quasi-isometries, replacing Teichmüller space by the curve complex, or considering different metrics instead.
In this talk, we present a different generalization, obtained in joint work with Carlos Serván, where we relax isometries to isometric embeddings. The result is that isometric embedding of Teichmüller spaces are, except for some low-dimensional special cases, induced by covering constructions. I.e., holomorphic maps obtained by pulling back complex structures along a branched covering of surfaces.