The Weil-Petersson Isometry Group

Let F = Fg,n be a surface of genus g with n punctures. We assume 3g − 3 + n > 1 and that (g, n) 6= (1, 2). The purpose of this paper is to prove, for the Weil-Petersson metric on Teichmuller space Tg,n, the analogue of Royden’s famous result [15] that every complex analytic isometry of Tg,0 with respect to the Teichmuller metric is induced by an element of the mapping class group. His proof involved a study of the local geometry of the cotangent bundle to Teichmuller space. Royden’s result was extended to general Tg,n by Earle-Kra [8], without any smoothness assumption on the isometry and with a stronger local result. They showed that if 2g + n > 4 and 2g + n > 4, and if f is an isometry from an open set U ⊂ Tg,n to Tg′,n′ , then Tg,n = Tg′,n′ and f is the restriction of an isometry induced by an element of the extended mapping class group. Later Ivanov [9] gave an alternative proof of Royden’s theorem based upon the asymptotic geometry of Teichmuller space and the result that the group of automorphisms of the curve complex C(F ) (see below) coincides with the mapping class group. The automorphism result was later extended to the cases of punctured surfaces of genus g ≤ 1 (with (g, n) 6= (1, 2)) by Korkmaz [10], and at the same time proved for general (g, n) 6= (1, 2) by Luo [11]. We prove