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Infinitesimal Liouville Distributions for Teichmüller Space

Published online by Cambridge University Press:  08 March 2004

Dragomir šarić
Dragomir šarić
Affiliation:
Dragomir šarić, USC Department of Mathematics, 1042 W. 36th Place, DRB 210 Los Angeles, CA 90089-1113, USA. E-mail: saric@math.usc.edu
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Abstract

We consider an arbitrary Riemann surface $X$, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space $G(\tilde{X})$ of geodesics of the universal covering $\tilde{X}$ of $X$. As we vary the Riemann surface structure, this gives an embedding from the Teichmüller space of $X$ into the Fréchet space of Hölder distributions on $G(\tilde{X})$. We show that the embedding is continuously differentiable. In particular, we obtain an explicit integral representation of the tangent map.

Keywords

Hölder distributiongeodesic currentLiouville measure30F6032G1546F99
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 88 , Issue 2 , March 2004 , pp. 436 - 454
Copyright
2004 London Mathematical Society

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