On complex extension of the Liouville map

TL;DR

This paper provides a new complex analysis proof that the Liouville map, which relates Teichmüller space to measures on geodesics, is real analytic, extending previous differentiability results to a broader context.

Contribution

It introduces a complex analysis approach to prove the real analyticity of the Liouville map, complementing existing geometric analysis methods.

Findings

Liouville map is real analytic for all Riemann surfaces.

Provides an alternative proof using complex analysis.

Extends differentiability results to a broader class of surfaces.

Abstract

The Liouville map assigns to each point in the Teichm\"uller space a positive Radon measure on the space of geodesics of the universal covering of the base Riemann surface. This construction which was introduced by Bonahon is valid for both finite and infinite Riemann surfaces. Bonahon and S\"ozen proved that the Liouville map is differentiable for closed Riemann surfaces and the second author extended this result to all other Riemann surfaces. Otal proved that the Liouville map is real analytic using an idea from the geometric analysis. The purpose of this note is to give another proof of Otal's result using a complex analysis approach.