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

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- by Itai Benjamini, Sergei Merenkov and Oded Schramm
- Proc. Amer. Math. Soc.
**132**(2004), 641-647 - DOI: https://doi.org/10.1090/S0002-9939-03-07147-8
- Published electronically: September 29, 2003
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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.## References

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## Bibliographic Information

**Itai Benjamini**- Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
- MR Author ID: 311800
- Email: itai@math.weizmann.ac.il
**Sergei Merenkov**- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: smerenko@math.purdue.edu
**Oded Schramm**- Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052
- Email: schramm@microsoft.com
- 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
- Communicated by: Jozef Dodziuk
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 641-647 - MSC (2000): Primary 14J15, 60J65
- DOI: https://doi.org/10.1090/S0002-9939-03-07147-8
- MathSciNet review: 2019938

Dedicated: In memory of Bob Brooks