Weil-Petersson Teichm\"{u}ller space III: dependence of Riemann mappings for Weil-Petersson curves

Abstract

The primary purpose of the paper is to study how a Riemann mapping depends on the corresponding Jordan curve. We are mainly concerned with those Jordan curves in the Weil-Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincar\'e metric. We endow the space of all normalized Weil-Petersson curves with a new real Hilbert manifold structure and show that it is topologically equivalent to the standard complex Hilbert manifold structure.