A counterexample in the classification of open Riemann surfaces

Kwon, Young K.

doi:10.1090/S0002-9939-1974-0330446-6

A counterexample in the classification of open Riemann surfaces
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Proc. Amer. Math. Soc. 42 (1974), 583-587 Request permission

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

Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
Corneliu Constantinescu and Aurel Cornea, Über den idealen Rand und einige seiner Anwendungen bei der Klassifikation der Riemannschen Flächen, Nagoya Math. J. 13 (1958), 169–233 (German). MR 96791
Young K. Kwon, Strict inclusion $0_{HB}<0_{HD}$ for all dimensions, Kyungpook Math. J. 13 (1973), 229–234. MR 341339

minimal but no HD-minimal

Mitsuru Nakai, A measure on the harmonic boundary of a Riemann surface, Nagoya Math. J. 17 (1960), 181–218. MR 123706
Leo Sario, Positive harmonic functions, Lectures on functions of a complex variable, University of Michigan Press, Ann Arbor, Mich., 1955, pp. 257–263. MR 0089923
L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. MR 0264064
Yukinari Tôki, On the classification of open Riemann surfaces, Osaka Math. J. 4 (1952), 191–201. MR 54054
Yukinari Tôki, On the examples in the classification of open Riemann surfaces. I, Osaka Math. J. 5 (1953), 267–280. MR 59375