The p-Weil–Petersson Teichmüller Space and the Quasiconformal Extension of Curves

Abstract

We consider the correspondence between the space of p-Weil–Petersson curves gamma on the plane and the p-Besov space of u=log gamma ' on the real line for p > 1. We prove that the variant of the Beurling–Ahlfors extension defined by using the heat kernel yields a holomorphic map for u on a domain of the p-Besov space to the space of p-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of p-integrable Beltrami coefficients to the p-Weil–Petersson Teichmüller space.