Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity Articles uri icon

publication date

  • May 2017

start page

  • 1097

end page

  • 1112


  • 7


  • 290

International Standard Serial Number (ISSN)

  • 0025-584X

Electronic International Standard Serial Number (EISSN)

  • 1522-2616


  • 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. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


  • poincare metric; quasi-isometry; gromov hyperbolic; linear isoperimetric inequality; green's function; voronoi diagram; gromov hyperbolicity; rough isometries; infinite type; isoperimetric-inequalities; manifolds; domains