Quadratic differentials and foliations on infinite Riemann surfaces (Q6610103)

A Riemann surface \(X\) is called infinite if its fundamental group is not finitely generated. A Riemann surface \(X\) is parabolic if it does not have a Green's function, that is, a harmonic function with a logarithmic singularity at a single point that is \(0\) at infinity.\N\NIn this paper, the author proves that a Riemann surface \(X\) is parabolic if and only if, for each integrable holomorphic quadratic differential \(\varphi\), the set of horizontal trajectories of \(\varphi\) that are crosscuts is of zero area. When \(X\) is parabolic, the author shows that each integrable holomorphic quadratic differential \(\varphi\) is approximated by a sequence of Jenkins-Strebel quadratic differentials with a single cylinder on \(X\), thus extending a result of Masur. The techniques used in the paper 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.