Kodai Mathematical Journal
- Kodai Math. J.
- Volume 33, Number 1 (2010), 99-115.
Existence of singular harmonic functions
An afforested surface W := <P, (Tn)nN, (σn)nN>, N being the set of positive integers, is an open Riemann surface consisting of three ingredients: a hyperbolic Riemann surface P called a plantation, a sequence (Tn)nN of hyperbolic Riemann surfaces Tn each of which is called a tree, and a sequence (σn)nN of slits σn called the roots of Tn contained commonly in P and Tn which are mutually disjoint and not accumulating in P. Then the surface W is formed by foresting trees Tn on the plantation P at the roots for all n N, or more precisely, by pasting surfaces Tn to P crosswise along slits σn for all n N. Let be the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions. One might feel that any afforested surface W := <P, (Tn)nN, (σn)nN> belongs to the family as far as its plantation P and all its trees Tn belong to . The aim of this paper is, contrary to this feeling, to maintain that this is not the case.
Kodai Math. J., Volume 33, Number 1 (2010), 99-115.
First available in Project Euclid: 6 April 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Nakai, Mitsuru; Segawa, Shigeo. Existence of singular harmonic functions. Kodai Math. J. 33 (2010), no. 1, 99--115. doi:10.2996/kmj/1270559160. https://projecteuclid.org/euclid.kmj/1270559160