Existence of singular harmonic functions

Kodai Mathematical Journal

Shigeo Segawa¹  Mitsuru Nakai² 

[1] Department of Mathematics Daido University;Department of Mathematics Nagoya Institute of Technology

关键词: afforested surface;    essentially positive;    hyperbolic;    Joukowski mapping;    parabolic;    Parreau decomposition;    Riemann surface;    quasibounded;    singular;    Wiener compactification;    Wiener (harmonic) boundary;   

DOI  :  10.2996/kmj/1270559160

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(10)An afforested surface W := , 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)n∈N of hyperbolic Riemann surfaces Tn each of which is called a tree, and a sequence (σn)n∈N 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 ${¥mathcal O}_s$ be the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions. One might feel that any afforested surface W := belongs to the family ${¥mathcal O}_s$ as far as its plantation P and all its trees Tn belong to ${¥mathcal O}_s$. The aim of this paper is, contrary to this feeling, to maintain that this is not the case.

【 授权许可】

Unknown