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Teichmüller space of a closed set in the Riemann sphere

Xinlong Dong, Arshiya Farhath. G, Sudeb Mitra

Abstract

The Teichmüller 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 Teichmüller spaces. In particular, we study the real-analyticity of Douady-Earle section for classical Teichmüller 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 Teichmüller 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.

Teichmüller space of a closed set in the Riemann sphere

Abstract

The Teichmüller 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 Teichmüller spaces. In particular, we study the real-analyticity of Douady-Earle section for classical Teichmüller 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 Teichmüller 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.
Paper Structure (22 sections, 13 theorems, 63 equations)

This paper contains 22 sections, 13 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Quasiconformal mappings
  3. Conformal naturality of Lieb isomorphism
  4. Teichmüller space of a closed set in $\widehat{\mathbb C}$
  5. $T(E)$ as a complex manifold
  6. Teichmüller metric on $T(E)$
  7. Lieb isomorphism
  8. Statement of Theorem A
  9. Proof of Theorem A
  10. Conformal naturality for $T(E)$
  11. Universal holomorphic motion of a closed set.
  12. Universal holomorphic motion of $E$.
  13. The case when $E = \widehat{\mathbb C}$
  14. Douady-Earle section
  15. Background
  16. ...and 7 more sections

Key Result

Proposition 1

(Lieb's isomorphism theorem.) For all $\mu$ and $\nu$ in $M(\mathbb C)$ we have $P_E(\mu) = P_E(\nu)$ if and only if $\widetilde{P}_E(\mu) = \widetilde{P}_E(\nu)$.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Corollary 1
  • proof
  • Definition 4
  • Definition 5
  • Theorem 1
  • Remark 1
  • ...and 23 more