# Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

Katsuhiko Matsuzaki, José M. Rodríguez

Research output: Contribution to journalArticle

### Abstract

We consider a planar Riemann surface R made of a non-compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in R caused by these points to be uniformly distributed in R in terms of Euclidean geometry. Then we associate a graph G with R by taking the Voronoi diagram for the uniformly distributed cusps and show that G represents certain geometric and analytic properties of R.

Original language English 1097-1112 16 Mathematische Nachrichten 290 7 https://doi.org/10.1002/mana.201400241 Published - 2017 May 1

### Fingerprint

Hyperbolicity
Cusp
Riemann Surface
Euclidean geometry
Voronoi Diagram
Set of points
Graph in graph theory

### Keywords

• Green's function
• Gromov hyperbolic
• linear isoperimetric inequality
• Poincaré metric
• quasi-isometry
• Voronoi diagram

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

In: Mathematische Nachrichten, Vol. 290, No. 7, 01.05.2017, p. 1097-1112.

Research output: Contribution to journalArticle

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