Parametrization of the p-Weil–Petersson Curves: Holomorphic Dependence

Abstract

Similar to the Bers simultaneous uniformization, the product of the p-Weil–Petersson Teichmüller spaces for p ge 1 provides the coordinates for the space of p-Weil–Petersson embeddings gamma of the real line {mathbb {R}} into the complex plane {mathbb {C}}. We prove the biholomorphic correspondence from this space to the p-Besov space of u=log gamma ' on {mathbb {R}} for p>1. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of p-Weil–Petersson curves. Specifically, it implies that the correspondence of the Riemann mapping parameters to the arc-length parameters preserving the images of curves is a homeomorphism with bi-real-analytic dependence of the change of parameters. This is analogous to the classical theorem of Coifman and Meyer for chord-arc curves.