Report on the teichmüller metric.

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.