Teichm\"uller space of a closed set in the Riemann sphere
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.
Cite
@article{arxiv.2603.05328,
title = {Teichm\"uller space of a closed set in the Riemann sphere},
author = {Xinlong Dong and Arshiya Farhath. G and Sudeb Mitra},
journal= {arXiv preprint arXiv:2603.05328},
year = {2026}
}