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A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature
| Venue: | Proc. Amer. Math. Soc |
| Citations: | 2 - 0 self |
BibTeX
@ARTICLE{Benjamini_anegative,
author = {Itai Benjamini and Sergei Merenkov and Oded Schramm},
title = {A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature},
journal = {Proc. Amer. Math. Soc},
year = {}
}
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Abstract
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 −ǫ times the area of disk, where ǫ> 0 is some constant. 1.
