Abstract
LetW be an open Riemann surface and\(\bar W\) ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ1 (resp.,\(\bar \Delta _1 \)) the minimal Martin boundary ofW (resp.,\(\bar W\)). For ζ ∈ Δ, let\(\nu _{\bar W} \)ζ be the (cardinal) number of the set of pionts\(\bar \varsigma \in \bar \Delta _1 \) which lie over ζ and\(\mathcal{M}_\varsigma \) the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following:\(\nu _{\bar W} \left( \varsigma \right) = \max _{M \in \mathcal{M}_\varsigma n} n_{\bar W} (M)\), where\(n_{\bar W} (M)\) is the number of components of π-1 M and π is the projection of\(\bar W\) ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.
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References
L. V. Ahlfors and L. Sario,Riemann Surfaces, Princeton, 1960.
H. Aikawa,Quasiadditivity of capacity and minimal thinness, Ann. Acad. Sci. Fenn. Ser. A I Math.18 (1993), 65–75.
J. Bliedtner and W. Hansen,Potential Theory, Springer, Berlin, 1986.
M. Brelot,On topologies and boundaries in potential theory, Lecture Notes in Math.175, Springer, Berlin, 1971.
C. Constantinescu and A. Cornea,Ideale Ränder Riemannscher Flächen, Springer, Berlin, 1969.
J. Deny,Un théorème sur les ensembles effilés, Ann. Univ. de Grenoble, Sect. Sci. Math. Phys.23 (1948), 139–142.
M. Essén,On minimal thinness, reduced functions and Green potentials, Proc. Edinburgh Math. Soc.36 (1992), 87–106.
O. Forster,Lectures on Riemann Surfaces, GTM 81, Springer, New York, 1981
M. Heins,Riemann surfaces of infinite genus, Ann. of Math.55 (1952), 296–317.
L. Helms,Introduction to Potential Theory, Wiley-Interscience, New York, 1969.
J. Lelong-Ferrand,Etude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup.66 (1949), 125–159.
H. Masaoka,Harmonic dimension of covering surfaces, II, Kodai Math. J.18 (1995), 487–493.
H. Masaoka,Criterion of a Wiener type for minimal thinness on covering surfaces, Proc. Japan Acad. Ser. A72 (1996), 154–156.
H. Masaoka and S. Segawa,Harmonic dimension of covering surfaces, Kodai Math. J.17 (1994), 351–359.
H. Masaoka and S. Segawa,Harmonic dimension of covering surfaces and minimal fine neighborhood, Osaka J. Math.34 (1997), 659–672.
L. Naïm,Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier7 (1957), 183–281.
L. Sario and M. Nakai,Classification theory of Riemann Surfaces, Springer, Berlin, 1970.
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Masaoka, H., Segawa, S. Martin boundary of unlimited covering surfaces. J. Anal. Math. 82, 55–72 (2000). https://doi.org/10.1007/BF02791221
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DOI: https://doi.org/10.1007/BF02791221