The finite Weil-Petersson diameter of Riemann space

Scott Wolpert
1977 Pacific Journal of Mathematics  
Let T g be the Teichmϋller space and R g the Riemann space of compact Riemann surfaces of genus g with g = 2. The space R g can be realized as the quotient of T g by a properly discontinuous group M g , the modular group. Various metrics have been defined for T g which are compatible with the standard topology for T g and induce quotient metrics for R g . Several authors have considered the Weil-Petersson metric for T g . A length estimate derived in a previous paper is summarized; combining
more » ... s 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 R g has finite diameter. A corollary is an estimate relating the Poincare length of the shortest closed geodesic of a compact Riemann surface to the Poincare diameter of the surface. For background material the reader is referred to the articles of L. Ahlfors [1] and L. Bers [3] and to the article of L. Bers [5] for a survey of related topics. T. C. Chu [7,8] and H. Masur [12] have obtained results related to ours. The author would like to thank Professor G. Kiremidjian for his assistance.
doi:10.2140/pjm.1977.70.281 fatcat:druseqm5nbcvxl5xvqe7khhbua