Simultaneous uniformization of chord-arc curves and BMO Teichmüller space

Abstract

This article surveys and develops the use of simultaneous uniformization for the study of chord-arc curves in the BMO Teichmüller space. The method of simultaneous uniformization provides a unified complex-analytic framework in which chord-arc curves are parametrized by their BMO embeddings and the logarithm of derivatives of these embeddings forms a biholomorphic image in the Banach space of BMO functions. We review this correspondence and its consequences, such as the relation to reparametrizations by strongly quasisymmetric homeomorphisms, in a rather self-contained manner in order to highlight the coherence of the approach.
The main new contribution of this exposition concerns the Cauchy transform of BMO functions on a chord-arc curve. We show that the Cauchy transform is expressed through the derivative of the biholomorphic map arising from simultaneous uniformization and consequently depends holomorphically on the variation of the chord-arc curve. This result connects classical singular integral operators with the complex structure of Teichmüller spaces and illustrates the effectiveness of the method. We also outline the parallel theory in the VMO Teichmüller space, which exhibits further structural properties.