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

Itai Benjamini; Sergei Merenkov; Oded Schramm

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

Itai Benjamini·Sergei Merenkov·Oded Schramm

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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. 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.

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Itai Benjamini, Sergei Merenkov, Oded Schramm. 2002-10-06. A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature. https://arxiv.org/abs/math/0209334

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