A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature

Abstract:

Consider a simply-connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic, and negative mean excess implies that the surface is hyperbolic. Teichmüller gave an example of a hyperbolic simply-connected Riemann surface whose mean excess is zero, disproving the first of these implications. We give an example of a simply-connected parabolic Riemann surface with negative mean excess, thus disproving the other part. We also construct an example of a complete, simply-connected, parabolic surface with nowhere positive curvature such that the integral of curvature in any disk about a fixed basepoint is less than $-\epsilon$ times the area of disk, where $\epsilon >0$ is some constant.