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

Xinlong Dong; Arshiya Farhath. G; Sudeb Mitra

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arXiv · 2603.05328

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

Xinlong Dong·Arshiya Farhath. G·Sudeb Mitra

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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.

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Xinlong Dong, Arshiya Farhath. G, Sudeb Mitra. 2026-03-05. Teichm\"uller space of a closed set in the Riemann sphere. https://arxiv.org/abs/2603.05328

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