  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. Teichmuller 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 the disk, where epsilon > 0 is some constant.
