Simultaneous uniformization of chord-arc curves and BMO Teichmüller space

Abstract: By using a method of simultaneous uniformization, we parametrize the space of chord-arc curves within the product of BMO Teichm\"uller spaces. This method provides a biholomorphic correspondence between such embeddings $\gamma$ onto chord-arc curves $\Gamma$ and $\log \gamma'$ in the Banach space of BMO functions. In our previous work, through this correspondence $\Lambda$, we have clarified the arguments of Teichm\"uller space theory for chord-arc curves, solved the problem of the discontinuity of the Riemann mapping parametrization of chord-arc curves, and simplified the proof for a real-analytic diffeomorphism onto the space of reparametrizations by strongly quasisymmetric homeomorphisms. In this paper, we first review these results by renewing the arguments in an effort to minimize reliance on other results. Then, we present a new application of simultaneous uniformization to the Cauchy transform of BMO functions on a chord-arc curve. We show that the Cauchy transform can be described by the derivative of the biholomorphic map $\Lambda$ and hence depends holomorphically on the variation of chord-arc curves. Finally, we organize and demonstrate the corresponding results for the VMO Teichm\"uller space.