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

@inproceedings{Benjamini2002ANA,
  title={A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature},
  author={Itai Benjamini and Sergei Merenkov and Oded Schramm},
  year={2002}
}
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… 

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References

SHOWING 1-10 OF 20 REFERENCES

Two-Dimensional Manifolds of Bounded Curvature

The theory of two-dimensional manifolds of bounded curvature is a generalization of two-dimensional Riemannian geometry. Formally a two-dimensional manifold of bounded curvature is a two-dimensional

Random walk on the Speiser graph of a Riemann surface

On considere la determination du type conforme d'une surface de recouvrement de la sphere de Riemann a n points. On montre comment definir une marche aleatoire sur les sommets du graphe de Speiser de

Random walks and electric networks

The goal will be to interpret Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the Starting Point when d ≥ 3, and to prove the theorem using techniques from classical electrical theory.

Quasiconformal mappings in the plane

I. Geometric Definition of a Quasiconformal Mapping.- to Chapter I.- 1. Topological Properties of Plane Sets.- 2. Conformal Mappings of Plane Domains.- 3. Definition of a Quasiconformal Mapping.- 4.

Potential Theory on Infinite Networks

Kirchhoff's laws.- Finite networks.- Currents and potentials withfinite energy.- Uniqueness and related topics.- Some examples and computations.- Royden's compactification.- Rough isometries.

Lecons sur les principes topologiques de la theorie des fonctions analytiques

Eindeutige Analytische Funktionen

Conformal Invariants: Topics in Geometric Function Theory

Intrinsic Geometry of Surfaces

Topics in geometric function theory