Projective structures on Riemann surfaces and Kleinian groups
type: Item
instance of: scholarly article
title: Projective structures on Riemann surfaces and Kleinian groups
zbMATH Open document ID: 0655.30028
author: Hiroshige Shiga
published in: Journal of Mathematics of Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Gamma\) be a Fuchsian universal covering group of a compact Riemann surface. Assume \(\Gamma\) acts on the upper half plane and let B(L,\(\Gamma)\) be the set of holomorphic functions \(\psi\) in the lower half plane which are holomorphic, which satisfy \(\psi (A(z))A'(z)^ 2=\psi (z)\) for all A in \(\Gamma\) and which have finite norm, \(\| \psi \| =\sup y^ 2| \psi (z)|\). If \(W_{\psi}\) is a holomorphic function in L whose Schwarzian derivative is equal to \(\psi\), then \(\psi\) determines a homomorphism \(\theta_{\psi}\) of \(\Gamma\) into PSL(2,\({\mathbb{C}})\) by the formula \[ w_{\psi}\circ A=\theta_{\psi}(A)\circ W_{\psi}\quad for\quad A\quad in\quad \Gamma. \] Let T(\(\Gamma)\) be the set of \(\psi\) for which \(W_{\psi}\) has a quasiconformal extension to the whole plane, let S(\(\Gamma)\) be the set of \(\psi\) such that \(W_{\psi}\) is schlicht in L and let K(\(\Gamma)\) be the set of \(\psi\) such that \(\Gamma^{\psi}=\theta_{\psi}(\Gamma)\) is a Kleinian group (with nonempty set of discontinuity in \({\mathbb{C}})\). The author proves the following theorems: Theorem 1. If \(\psi_ 0\) and \(\psi_ 1\) lie in a single component of Int(K(\(\Gamma)\)), then \(\Gamma^{\psi_ 1}\) and \(\Gamma^{\psi_ 2}\) are quasiconformally equivalent, that is, there exists a quasiconformal selfmapping w of \({\hat {\mathbb{C}}}\) such that \(w\circ A\circ w^{- 1}=\theta_{\psi_ 1}\circ \theta_{\psi_ 0}(A)\) for all A in \(\Gamma_{\psi_ 0}\). Moreover, if \(\| \psi_ 0-\psi_ 1\|\) is sufficiently small, then the quasiconformal selfmapping can be taken to have harmonic Beltrami coefficient. Theorem 2. For every \(\psi\) in Int(K(\(\Gamma)\)-T(\(\Gamma)\)), the function \(W_{\psi}\) is not a covering mapping on L. Consequently, \(W_{\psi}(L)={\hat {\mathbb{C}}}\). By definition, a b-group G is a group with only one simply connected invariant component \(\Delta\) in its set of discontinuity. Using the same formalism, one can construct a homomorphism \(\chi_{\psi}\) of G …
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 4069280
zbMATH Keywords: projective structures
zbMATH Keywords: deformations.
zbMATH Keywords: Kleinian group
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520657
OpenAlex ID: W1555228064
DOI: 10.1215/KJM/1250520657
Recommended article: Projective structures on a Riemann surface
Recommended article: Q4217346
Recommended article: Notes on projective structures and Kleinian groups
Recommended article: Projective structures and projective bundles over compact Riemann surfaces
Recommended article: Q4719524
Recommended article: Projections in moduli spaces of the Kleinian groups
Recommended article: Projective structures on a Riemann surface. III
Recommended article: On singular projective structures on Riemann surfaces
Recommended article: Projective structures on Riemann surface and natural differential operators
Recommended article: PROJECTIVE STRUCTURES AND HIGGS BUNDLES ON SURFACES
description: scientific article; zbMATH DE number 4069280