Advanced Studies in Pure Mathematics

Hyperbolic Riemann surfaces without unbounded positive harmonic functions

Hiroaki Masaoka and Shigeo Segawa

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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

Potential Theory in Matsue, H. Aikawa, T. Kumagai, Y. Mizuta and N. Suzuki, eds. (Tokyo: Mathematical Society of Japan, 2006), 227-232

Received: 25 March 2005
Revised: 7 June 2005
First available in Project Euclid: 16 December 2018

Permanent link to this document euclid.aspm/1544999693

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F15: Harmonic functions on Riemann surfaces 30F20: Classification theory of Riemann surfaces 30F25: Ideal boundary theory 31C35: Martin boundary theory [See also 60J50]

hyperbolic Riemann surface positive harmonic function Martin boundary harmonic measure


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.

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