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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
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?

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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