A Negative Answer to Nevanlinna’s Type Question and a Parabolic Surface with a Lot of Negative Curvature

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 −ǫ times the area of disk, where ǫ > 0 is some constant.