On the derivatives of the Liouville currents

TL;DR

This paper explores the derivatives of the Liouville map in Teichmüller and Quasi-Fuchsian spaces, providing geometric expressions for derivatives along earthquake paths, enhancing understanding of the map's analytic properties.

Contribution

It offers a new geometric formula for the derivative of the Liouville map along earthquake paths, connecting Teichmüller theory with complex analysis.

Findings

Derived a geometric expression for the Liouville map's derivative

Extended the Liouville map to a holomorphic map in Quasi-Fuchsian space

Connected earthquake paths with the analytic properties of the Liouville map

Abstract

The Liouville map, introduced by Bonahon, assigns to each point in the Teichm\"uller space a natural Radon measure on the space of geodesics of the base surface. The Liouville map is real analytic and it even extends to a holomorphic map of a neighborhood of the Teichm\"uller space in the Quasi-Fuchsian space of an arbitrary conformally hyperbolic Riemann surface. The earthquake paths and by their extension quake-bends, introduced by Thurston, are particularly nice real-analytic and holomorphic paths in the Teichm\"uller and the Quasi-Fuchsian space, respectively. We find a geometric expression for the derivative of the Liouville map along earthquake paths.