Isometric embeddings and totally geodesic submanifolds of Teichmüller spaces

Frederik Benirschke, University of Chicago

Classical results by Royden, Earle, and Kra imply that the biholomorphism group of Teichmüller space, the isometry group of the Teichmüller metric, 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.

In this talk, we present a generalization, obtained in joint work with Carlos Serván, where we relax isometries to isometric embeddings. The main result is that isometric embeddings of Teichmüller spaces are coverings constructions, except for some low-dimensional special cases. In other words: Isometric embeddings are induced by branched coverings of the underlying surfaces.

Time permitting, we explain how our techniques can be used to rule out the existence of certain totally geodesic submanifolds.

QNC 2501