Nakai, Mitsuru; Segawa, Shigeo; Tada, Toshimasa Surfaces carrying no singular functions. (English) Zbl 1193.30058 Proc. Japan Acad., Ser. A 85, No. 10, 163-166 (2009). Let \(X\) and \(Y\) be two Riemann surfaces, and let \(\gamma\) be a slit commonly contained in both. A Riemann surface \(Z= X\uplus Y\) is obtained by pasting \(X\setminus\gamma\) to \(Y\setminus\gamma\) crosswise along \(\gamma\). For \(j\in J= \{1,2,3,\dots,m\}\), let \(W_j\) be an open Riemann surface. If \(J'= \{j_1,j_2,\dots, j_m\}\) is a permutation of \(J\) and \(Z_1= W_{j_1}\uplus W_{j_2}\) for a common slit \(\gamma_{j_1}\) in \(W_{j_1}\), set \(Z_2= Z_1\uplus W_{j_3}\) for a common slit \(\gamma_{j_2}\) in \(Z_1\) and \(W_{j_3}\). Continuing, the surface \(Z_{m-1}= Z_{m-2}\uplus W_{j_m}\) is obtained. Neglecting how the permutation and the sequence of pasting slits are chosen, a resulting surface, called the united surface of the \(W_j\), is formed. The authors show that there is a canonical isomorphism between the space of harmonic functions on the united surface and the space of harmonic functions on the bunched surface \(\bigcup_{j\in J} W_j\), assuming all the \(W_j\) are hyperbolic surfaces. An application of this result gives a sufficient condition for an afforested surface to belong to the class \(O_s\) of hyperbolic Riemann surfaces which have no nonzero singular harmonic functions for which its plantation and trees on it are all in \(O_s\). Reviewer: L. R. Sons (DeKalb) Cited in 1 Document MSC: 30F20 Classification theory of Riemann surfaces 30F15 Harmonic functions on Riemann surfaces 30F25 Ideal boundary theory for Riemann surfaces Keywords:Riemann surface; afforested surface; hyperbolic surface PDFBibTeX XMLCite \textit{M. Nakai} et al., Proc. Japan Acad., Ser. A 85, No. 10, 163--166 (2009; Zbl 1193.30058) Full Text: DOI References: [1] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen , Ergebnisse der Mathematik und ihre Grenzgebiete, Band 32, Springer, Berlin, 1963. · Zbl 0112.30801 [2] F.-Y. Maeda, Dirichlet integrals on harmonic spaces , Lecture Notes in Math., 803, Springer, Berlin, 1980. · Zbl 0426.31001 [3] H. Masaoka and S. Segawa, On several classes of harmonic functions on a hyperbolic Riemann surface, in Complex analysis and its applications , 289-294, Osaka Munic. Univ. Press, Osaka, 2008. · Zbl 1190.30028 [4] M. Nakai and S. Segawa, Types of afforested surfaces, Kodai Math. J. 32 (2009), no. 1, 109-116. · Zbl 1162.30027 [5] M. Nakai and S. Segawa, Existence of singular harmonic functions, Kodai Math. J. (to appear). · Zbl 1192.30014 [6] M. Nakai and T. Tada, Monotoneity and homogeneity of Picard dimensions for signed radial densities, Hokkaido Math. J. 26 (1997), no. 2, 253-296. · Zbl 0878.31004 [7] B. Rodin and L. Sario, Principal functions , University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1968. · Zbl 0159.10701 [8] L. Sario and M. Nakai, Classification theory of Riemann surfaces , Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 164, Springer, New York, 1970. · Zbl 0199.40603 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.