Infinitesimal Liouville currents, cross-ratios and intersection numbers

TL;DR

This paper establishes a connection between cross-ratio functions on the circle at infinity of a surface's universal cover and the Weil-Petersson scalar product, generalizing classical intersection numbers to tangent vectors in Teichmüller space.

Contribution

It introduces a new geometric intersection number for cross-ratio functions associated with tangent vectors in Teichmüller space, linking it to the Weil-Petersson scalar product.

Findings

Defined a geometric intersection number for regular cross-ratio functions.

Proved the intersection number equals the Weil-Petersson scalar product for tangent vectors.

Extended classical intersection concepts to a broader setting in Teichmüller theory.

Abstract

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichm\"uller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichm\"uller space, we show that two such cross-ratio functions have a well-defined geometric intersection number, and that this intersection number is equal to the Weil-Petersson scalar product of the corresponding vectors.