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A counterexample in the classification of open Riemann surfaces


Author: Young K. Kwon
Journal: Proc. Amer. Math. Soc. 42 (1974), 583-587
MSC: Primary 30A48
DOI: https://doi.org/10.1090/S0002-9939-1974-0330446-6
MathSciNet review: 0330446
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Abstract: An HD-function (harmonic and Dirichlet-finite) $ \omega $ on a Riemann surface R is called HD-minimal if $ \omega > 0$ and every HD-function $ \omega '$ with $ 0 \leqq \omega ' \leqq \omega $ reduces to a constant multiple of $ \omega $. An $ H{D^ \sim }$-function is the limit of a decreasing sequence of positive HD-functions and $ H{D^\sim}$-minimality is defined as in HD-functions. The purpose of the present note is to answer in the affirmative the open question: Does there exist a Riemann surface which carries an $ HD^\sim $-minimal function but no HD-minimal functions?


References [Enhancements On Off] (What's this?)

  • [1] L. Ahlfors and L. Sario, Riemann surfaces, Princeton Math. Series, no. 26, Princeton Univ. Press, Princeton, N.J., 1960. MR 22 #5729. MR 0114911 (22:5729)
  • [2] C. Constantinescu and A. Cornea, Über den idealen Rand und einige seiner Anwendungen bei der Klassifikation der Riemannschen Flächen, Nagoya Math. J. 13 (1958), 169-233. MR 20 #3273. MR 0096791 (20:3273)
  • [3] Y. K. Kwon, Strict inclusion $ {O_{HB}} < {O_{HD}}$ for all dimensions, Kyungpook Math. J. (to appear). MR 0341339 (49:6090)
  • [4] -, $ H{D^ \sim }$-minimal but no HD-minimal, Pacific J. Math. (to appear).
  • [5] M. Nakai, A measure on the harmonic boundary of a Riemann surface, Nagoya Math. J. 17 (1960), 181-218. MR 23 #A1028. MR 0123706 (23:A1028)
  • [6] L. Sario, Positive harmonic functions. Lectures on functions of a complex variable, Univ. Michigan Press, Ann Arbor, Mich., 1955, pp. 257-263. MR 19, 739. MR 0089923 (19:739b)
  • [7] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der Math. Wissenschaften, Band 164, Springer-Verlag, New York and Berlin, 1970, 446 pp. MR 41 #8660. MR 0264064 (41:8660)
  • [8] Y. Tôki, On the classification of open Riemann surfaces, Osaka Math. J. 4 (1952), 191-201. MR 14, 864. MR 0054054 (14:864a)
  • [9] -, On examples in the classification of Riemann surfaces. I, Osaka Math. J. 5 (1953), 267-280. MR 15, 519. MR 0059375 (15:519c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0330446-6
Keywords: HD-function, HD-minimal functions, $ HD^\sim$-function, $ HD^\sim$-minimal function, Dirichlet integral, Royden's compactification, Wiener's compactification, harmonic boundary, harmonic kernel, Riemannian n-manifold
Article copyright: © Copyright 1974 American Mathematical Society

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