## Advanced Studies in Pure Mathematics

### Hyperbolic Riemann surfaces without unbounded positive harmonic functions

#### Abstract

Let $R$ be an open Riemann surface with Green's functions. It is proved that there exist no unbounded positive harmonic functions on $R$ if and only if the minimal Martin boundary of $R$ consists of finitely many points with positive harmonic measure.

#### Article information

Dates
Revised: 7 June 2005
First available in Project Euclid: 16 December 2018

https://projecteuclid.org/ euclid.aspm/1544999693

Digital Object Identifier
doi:10.2969/aspm/04410227

Mathematical Reviews number (MathSciNet)
MR2277836

Zentralblatt MATH identifier
1121.31006

#### Citation

Masaoka, Hiroaki; Segawa, Shigeo. Hyperbolic Riemann surfaces without unbounded positive harmonic functions. Potential Theory in Matsue, 227--232, Mathematical Society of Japan, Tokyo, Japan, 2006. doi:10.2969/aspm/04410227. https://projecteuclid.org/euclid.aspm/1544999693