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. 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2002
 arXiv:
 arXiv:math/0209334
 Bibcode:
 2002math......9334B
 Keywords:

 Mathematics  Complex Variables;
 30F45
 EPrint:
 7 pages, 2 figures, LaTex