Intersection numbers between horizontal foliations of quadratic differentials

TL;DR

This paper proves the finiteness and continuity of intersection numbers between horizontal foliations of quadratic differentials on Riemann surfaces, revealing that Jenkins-Strebel differentials are not dense in certain cases.

Contribution

It establishes the joint continuity of intersection numbers and shows Jenkins-Strebel differentials are not dense in the space of finite-area quadratic differentials on non-parabolic surfaces.

Findings

Intersection numbers are finite for all pairs of quadratic differentials.

Intersection number is jointly continuous in the $L^1$-norm.

Jenkins-Strebel differentials are not dense in the space when the surface is not parabolic.

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 $L^1$-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.