Masaoka, Hiroaki; Segawa, Shigeo Martin boundary of unlimited covering surfaces. (English) Zbl 0965.30014 J. Anal. Math. 82, 55-72 (2000). From the authors’ abstract: Let \(W\) be an open Riemann surface and \(\widetilde{W}\) a \(p\)-sheeted \((1 < p < \infty)\) unlimited covering surface of \(W\). Denote by \(\Delta_1\) (resp., \(\widetilde{\Delta}_1\)) the minimal Martin boundary of \(W\) (resp., \(\widetilde{W}\)). For \(\zeta \in \Delta_1\), let \(\nu_{\widetilde{W}}(\zeta)\) be the (cardinal) number of the set of points \(\widetilde{\zeta} \in \widetilde{\Delta}_1\) which lie over \(\zeta\) and \(\mathcal{M}_{\zeta}\) the class of open connected subsets \(M\) of \(W\) such that \(M \cup \{\zeta\}\) is a minimal fine neighborhood of \(\zeta\). Our main result is the following: \(\nu_{\widetilde{W}}(\zeta) = \max_{M \in \mathcal{M}_{\zeta}} n_{\widetilde{W}}(M)\), where \(n_{\widetilde{W}}(M)\) is the number of components of \(\pi^{-1}(M)\) and \(\pi\) is the projection of \(\widetilde{W}\) onto \(W\). Moreover, some applications of the above results are discussed when \(W\) is the unit disc. Reviewer: H.Köditz (Hannover) Cited in 1 Document MSC: 30F25 Ideal boundary theory for Riemann surfaces 31C35 Martin boundary theory Keywords:potential theory; Riemann surfaces PDFBibTeX XMLCite \textit{H. Masaoka} and \textit{S. Segawa}, J. Anal. Math. 82, 55--72 (2000; Zbl 0965.30014) Full Text: DOI References: [1] L. V. Ahlfors and L. Sario,Riemann Surfaces, Princeton, 1960. · Zbl 0196.33801 [2] Aikawa, H., Quasiadditivity of capacity and minimal thinness, Ann. Acad. Sci. Fenn. Ser. A I Math., 18, 65-75 (1993) · Zbl 0795.31005 [3] Bliedtner, J.; Hansen, W., Potential Theory (1986), Berlin: Springer, Berlin · Zbl 0706.31001 [4] Brelot, M., On topologies and boundaries in potential theory (1971), Berlin: Springer, Berlin · Zbl 0222.31014 [5] Constantinescu, C.; Cornea, A., Ideale Ränder Riemannscher Flächen (1969), Berlin: Springer, Berlin · Zbl 0112.30801 [6] Deny, J., Un théorème sur les ensembles effilés, Ann. Univ. de Grenoble, Sect. Sci. Math. Phys., 23, 139-142 (1948) · Zbl 0030.05602 [7] Essén, M., On minimal thinness, reduced functions and Green potentials, Proc. Edinburgh Math. Soc., 36, 87-106 (1992) · Zbl 0789.31001 [8] Forster, O., Lectures on Riemann Surfaces (1981), New York: Springer, New York · Zbl 0475.30002 [9] Heins, M., Riemann surfaces of infinite genus, Ann. of Math., 55, 296-317 (1952) · Zbl 0046.08702 [10] Helms, L., Introduction to Potential Theory (1969), New York: Wiley-Interscience, New York · Zbl 0188.17203 [11] Lelong-Ferrand, J., Etude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66, 125-159 (1949) · Zbl 0033.37301 [12] Masaoka, H., Harmonic dimension of covering surfaces, II, Kodai Math. J., 18, 487-493 (1995) · Zbl 0858.30033 [13] Masaoka, H., Criterion of a Wiener type for minimal thinness on covering surfaces, Proc. Japan Acad. Ser. A, 72, 154-156 (1996) · Zbl 0866.31006 [14] Masaoka, H.; Segawa, S., Harmonic dimension of covering surfaces, Kodai Math. J., 17, 351-359 (1994) · Zbl 0811.30028 [15] Masaoka, H.; Segawa, S., Harmonic dimension of covering surfaces and minimal fine neighborhood, Osaka J. Math., 34, 659-672 (1997) · Zbl 0968.30020 [16] Naïm, L., Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, 7, 183-281 (1957) · Zbl 0086.30603 [17] Sario, L.; Nakai, M., Classification theory of Riemann Surfaces (1970), Berlin: Springer, Berlin · 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.