Noncompleteness of the weil-petersson metric for teichmüller space

Abstract

Let Tg be the Teichmiiller space of a compact Riemann surface R of genus g with g ≧ 2. In the present paper it is shown thatthe Weil-Petersson length of a large class of rays is finite, deduced that the metric is not complete and indicated how the proof can be extended to the Teichmiiller space of an arbitrary finitely generated Fuchsian group of the first kind. The proof is carried out by estimating the Weil-Petersson length of Teichmiiller geodesic rays in directions corresponding to a certain class of quadratic differentials. © 1975 Pacific Journal of Mathematics.