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

TL;DR

This paper explores the complex structure of the space of chord-arc curves via BMO Teichmüller spaces, proving a biholomorphic correspondence and analyzing the Riemann mapping parametrization, ultimately resolving a longstanding conjecture about its discontinuity.

Contribution

It establishes a biholomorphic homeomorphism between chord-arc curves and BMO functions, and clarifies the analytic dependence of Riemann mapping parametrization, solving a 1990 conjecture.

Findings

Logarithm of the derivative provides a biholomorphic map into BMO functions.

The Riemann mapping parametrization depends discontinuously on arc-length.

The space of chord-arc curves can be embedded into BMO Teichmüller spaces with complex structures.

Abstract

We consider the space of chord-arc curves on the plane passing through the infinity with their parametrization $\gamma$ on the real line, and embed this space into the product of the BMO Teichm\"uller spaces. The fundamental theorem we prove about this representation is that $\log \gamma'$ also gives 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 a conjecture of Katznelson, Nag, and Sullivan in 1990 by showing that this dependence is not continuous.