Quadratic differentials and foliations on infinite Riemann surfaces

Abstract

We prove that an infinite Riemann surface X is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are crosscuts is of zero measure. Then we establish the density of the Jenkins–Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff’s formula for the Teichmüller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard–Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials.