Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Abstract

Let X X and Y Y be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of X X onto the Teichmüller space of Y Y is induced by a quasiconformal homeomorphism of X X onto Y Y . These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.