Vol. 70, No. 1, 1977

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The finite Weil-Petersson diameter of Riemann space

Scott Andrew Wolpert

Vol. 70 (1977), No. 1, 281–288

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.

Mathematical Subject Classification 2000
Primary: 32G15
Received: 23 June 1976
Published: 1 May 1977
Scott Andrew Wolpert