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

Authors: Itai Benjamini, Sergei Merenkov and Oded Schramm
Journal: Proc. Amer. Math. Soc. 132 (2004), 641-647
MSC (2000): Primary 14J15, 60J65
Published electronically: September 29, 2003
MathSciNet review: 2019938
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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.

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Additional Information

Itai Benjamini
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

Sergei Merenkov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Oded Schramm
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052

Received by editor(s): October 17, 2002
Published electronically: September 29, 2003
Additional Notes: The research of the second author was supported by NSF grant DMS-0072197
Dedicated: In memory of Bob Brooks
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2003 American Mathematical Society

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