Differentiability of the Liouville Map via Geodesic Currents

Abstract

For a conformally hyperbolic Riemann surface, the Teichmüller space is the space of quasiconformal maps factored by an equivalence relation, and it is a complex Banach manifold. The space of geodesic currents endowed with the uniform weak* topology is a subset of a Fréchet space of Hölder distributions. We introduce an appropriate topology on the space of Hölder distributions and this new topology coincides with the uniform weak* topology on the space of geodesic currents. The Liouville map of the Teichmüller space becomes differentiable in the Fréchet sense. In particular, the derivative of Liouville currents exists and belongs to the space of Hölder distributions, and the tangent map of the Liouville map is continuous and linear. The elements of the Teichmüller space can be represented by earthquake maps. Since an earthquake path is a differentiable path in the Teichmüller space, then the image of an earthquake path under the Liouville map is a differentiable path in the space of Hölder distributions. We compute the image of the tangent vector to an earthquake path in the space of Hölder distributions