Schiffer variation of complex structure and coordinates for Teichmuller spaces

Nag, Subashis (1985) Schiffer variation of complex structure and coordinates for Teichmuller spaces Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 94 (2-3). pp. 111-122. ISSN 0253-4142

PDF - Publisher Version

Official URL:

Related URL:


Schiffer variation of complex structure on a Riemann surface X0 is achieved by punching out a parametric disc D̅ from X0 and replacing it by another Jordan domain whose boundary curve is a holomorphic image of ∂D̅. This change of structure depends on a complex parameter e which determines the holomorphic mapping function around. It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmuller space T(X0), (or reduced Teichmuller space T#(X0)). For compact X0 this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates. Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result. Theorems 1 and 2 below take care of all the finitedimensional Teichmuller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensional T(X0) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.

Item Type:Article
Source:Copyright of this article belongs to Indian Academy of Sciences.
Keywords:Riemann Surfaces; Teichmuller Spaces; Quasiconformal Mappings
ID Code:33318
Deposited On:21 Mar 2011 14:03
Last Modified:17 May 2016 16:10

Repository Staff Only: item control page