Dragomir \v{S}ari\'{c}

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.

2000 Mathematical Subject Classification: 30F60, 32G15 (primary), 46F99 (secondary).

Keywords: Hölder distribution, geodesic current, Liouville measure.

E-mail:
saric@math.usc.edu