Parametrization of the $p$-Weil-Petersson curves: holomorphic dependence

Abstract

We prove the biholomorphic correspondence from the space of p-Weil-Petersson curves γ on the plane identified with the product of the p-Weil-Petersson Teichmüller spaces to the p-Besov space of u = log γ on the real line for p ≥ 2. From this result, several consequences follow immediately which clarify the analytic structures of parameter spaces of p-Weil-Petersson curves. In particular, generalizing the case of p = 2, the correspondence keeping the image of curves from the real-analytic submanifold for arc-length parametrizations to the complex-analytic submanifold for Riemann mapping parametrizations is a homeomorphism with real-analytic dependence of change of parameters. 1/2 R of real-valued functions. Then, Shen and Tang [33] regarded H 1/2 R as a new parameter space for T 2 which is real-analytically equivalent to the original complex Hilbert structure. In this work, they considered the Weil-Petersson class W 2 on

The VMO-Teichmüller space and the variant of Beurling–Ahlfors extension by heat kernel

“…Then, by the argument of simultaneous uniformization due to Bers, we see the following fact. This method for the investigation of embedded curves first appeared in [33].…”