Intersection numbers between horizontal foliations of quadratic differentials

2025/06/17 by Saric, Dragomir, Shima, Taro · 1 citation
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2506.14943

Abstract

We establish that the intersection number between the horizontal foliations of any two finite-area holomorphic quadratic differentials on an arbitrary Riemann surface is finite. Our main result shows that the intersection number is jointly continuous in the L¹-norm on the quadratic differentials. A corollary is that the Jenkins-Strebel differentials are not dense in the space of all finite-area holomorphic quadratic differentials when the infinite Riemann surface is not parabolic.

Citations

• Quadratic differentials and function theory on Riemann surfaces
• Quadratic differentials and foliations on infinite Riemann surfaces
• The type problem for Riemann surfaces via Fenchel-Nielsen parameters
• The Heights Theorem for infinite Riemann surfaces
• Train tracks and measured laminations on infinite surfaces

Cited by

• Teichmüller extremal maps on infinite Riemann surfaces

Related

• Quadratic differentials and foliations on infinite Riemann surfaces
• Quadratic differentials and function theory on Riemann surfaces
• The Heights Theorem for infinite Riemann surfaces
• Horizontal strips and spaces of quadratic differentials
• Holomorphic quadratic differentials in Teichmüller theory