Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

Katsuhiko Matsuzaki, José M. Rodríguez*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 languageEnglish
Pages (from-to)1097-1112
Number of pages16
JournalMathematische Nachrichten
Issue number7
Publication statusPublished - 2017 May


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

ASJC Scopus subject areas

  • Mathematics(all)


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