Weil–Petersson Teichmüller space III: dependence of Riemann mappings for Weil–Petersson curves

Abstract

The classical Riemann mapping theorem implies that there exists a so-called Riemann mapping which takes the upper half plane onto the left domain bounded by a Jordan curve in the extended complex plane. The primary purpose of the paper is to study the basic problem: how does a Riemann mapping depend 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 Poincare metric. After giving a geometric characterization of a Weil–Petersson curve, we endow the space of all normalized Weil–Petersson curves with a new real Hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex Hilbert manifold structure, which implies that an appropriately chosen Riemann mapping depends continuously on a Weil–Petersson curve (and vice versa). This can be considered as the first result about the continuous dependence of Riemann mappings on non-smooth Jordan curves.