Premium
Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity
Author(s) -
Matsuzaki Katsuhiko,
Rodríguez José M.
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400241
Subject(s) - mathematics , riemann surface , planar , voronoi diagram , euclidean geometry , simply connected space , plane (geometry) , graph , planar graph , surface (topology) , riemann hypothesis , mathematical analysis , geometry , pure mathematics , combinatorics , computer graphics (images) , computer science
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 .