Report on the Teichmüller Metric
Abstract
Let T^{g} be the Teichmüller space of compact Riemann surfaces of genus g. Then T^{g} is the space of conformal structures on a fixed surface W modulo equivalence under conformal maps homotopic to the identity. The Teichmüller modular group [unk] is the group of all orientation preserving homeomorphisms of W onto itself modulo those which are homotopic to the identity. Each element of [unk] induces a biholomorphic map of T^{g} onto itself, and the present note outlines a proof of the converse statement: Every biholomorphic map of T^{g} onto itself is induced by an element of [unk]. It is first shown that every isometry of T^{g} with the Teichmüller metric arises from an element of [unk]. The Teichmüller metric is then shown to be the Kobayashi metric for T^{g} and hence invariant under biholomorphic maps.
 Publication:

Proceedings of the National Academy of Science
 Pub Date:
 March 1970
 DOI:
 10.1073/pnas.65.3.497
 Bibcode:
 1970PNAS...65..497R