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

Abstract

AbstractSimilar to the Bers simultaneous uniformization, the product of thep-Weil–Petersson Teichmüller spaces for$$p \ge 1$$p≥1provides the coordinates for the space ofp-Weil–Petersson embeddings$$\gamma $$γof the real line$${\mathbb {R}}$$Rinto the complex plane$${\mathbb {C}}$$C. We prove the biholomorphic correspondence from this space to thep-Besov space of$$u=\log \gamma '$$u=logγ′on$${\mathbb {R}}$$Rfor$$p>1$$p>1. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces ofp-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.