English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Many classical objects on a surface  S  can be interpreted as cross‐ratio functions on the circle at infinity of the universal covering  S̃ . 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 Teichmüller 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 Teichmüller 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 Riemannian product of the corresponding vectors.

Infinitesimal Liouville currents, cross‐ratios and intersection numbers