Let Tg be the Teichmüller
space and Rg the Riemann space of compact Riemann surfaces of genus g with g ≧ 2.
The space Rg can be realized as the quotient of Tg by a properly discontinuous group
Mg, the modular group. Various metrics have been defined for Tg which are
compatible with the standard topology for Tg and induce quotient metrics for Rg.
Several authors have considered the Weil-Petersson metric for Tg. A length estimate
derived in a previous paper is summarized; combining this with the Ahlfors Schwarz
lemma, an estimate of N. Halpern and L. Keen, and an additional argument
shows that the Weil-Petersson quotient metric for Rg has finite diameter. A
corollary is an estimate relating the Poincaré length of the shortest closed
geodesic of a compact Riemann surface to the Poincaré diameter of the
surface.
