Function theoretic characterizations of Weil–Petersson curves

Abstract

The Weil–Petersson class is the closure of the smooth closed curves in the Weil–Petersson metric on universal Teichmüller space defined by Takhtajan and Teo. We give some new characterizations of this class of curves and some new proofs of previously known characterizations. In particular, we give a new, more geometric characterization of the conformal weldings of such curves and characterize the curves themselves in terms of Peter Jones’s $\beta$-numbers.