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

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