Simultaneous uniformization of chord-arc curves and BMO Teichm\"uller space

TL;DR

This paper explores the use of simultaneous uniformization to study chord-arc curves within the BMO Teichmüller space, linking complex analysis, singular integrals, and geometric function theory.

Contribution

It introduces a new connection between the Cauchy transform of BMO functions on chord-arc curves and the derivatives of uniformizing maps, advancing the understanding of Teichmüller space structures.

Findings

Cauchy transform expressed via derivatives of uniformizing maps

Holomorphic dependence of transforms on curve variations

Parallel theory developed for VMO Teichmüller space

Abstract

This article surveys and develops the use of simultaneous uniformization for the study of chord-arc curves in the BMO Teichm\"uller 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 form 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 Teichm\"uller spaces and illustrates the effectiveness of the method. We also outline the parallel theory in the VMO Teichm\"uller space, which exhibits further structural properties.