Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

Katsuhiko Matsuzaki, José M. Rodríguez

    Research output: Contribution to journalArticle


    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 1



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

    ASJC Scopus subject areas

    • Mathematics(all)

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