Teichm\"uller space of a closed set in the Riemann sphere

TL;DR

This paper investigates the complex structure and naturality properties of Teichmüller spaces of closed sets in the Riemann sphere, focusing on the Douady-Earle section and real-analyticity of holomorphic motions.

Contribution

It demonstrates the conformal naturality of Lieb isomorphism and studies the real-analyticity of the Douady-Earle section in classical Teichmüller spaces, providing explicit examples and new results on holomorphic motions.

Findings

Lieb isomorphism is conformally natural.

Douady-Earle section is real-analytic for classical Teichmüller spaces.

Family of Jordan curves varies real-analytically with holomorphic marked points.

Abstract

The Teichm\"uller space of a closed set in the Riemann sphere is a simply connected complex Banach manifold. Its complex structure follows from Lieb isomorphism. In this paper, we show the conformal naturality of Lieb isomorphism. We then study Douady-Earle section for these Teichm\"uller spaces. In particular, we study the real-analyticity of Douady-Earle section for classical Teichm\"uller spaces. We give two explicit examples of maximal holomorphic motions over simply connected complex Banach manifolds. As an application of the real-analyticity of the Douady-Earle section for the classical Teichm\"uller spaces of Riemann surfaces, we prove a new result showing that a family of Jordan curves varies real-analytically over a simply connected complex Banach manifold and as quasiconformal images of the one at the basepoint, provided that a finite number of marked points on the Jordan curves vary holomorphically over the same parameter space.