BMO embeddings, chord-arc curves, and Riemann mapping parametrization

Abstract

We consider the space of chord-arc curves on the plane passing through infinity with their parametrization γ defined on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that γ↦log⁡γ′ is a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve the conjecture of Katznelson, Nag, and Sullivan by showing that this dependence is not continuous.