Horizontal foliations of integrable holomorphic quadratic differentials on infinite Riemann surfaces

Let X be an infinite Riemann surface. We prove that the Brownian motion on X is recurrent iff a.e. horizontal leaf of every integrable holomorphic quadratic differential on X is recurrent. When X is such, we prove that the holomorphic quadratic differentials with single horizontal cylinders (the Jenkins-Strebel differentials) are dense among all integrable holomorphic quadratic differentials in the L^1-norm. We also extend Kerckhoff’s formula for the Teichmuller distance in terms of the extremal lengths of simple curves for such Riemann surfaces.