A Counterexample in the Classification of Open Riemann Surfaces - Kwon, Young Koan (Institutional Knowledge at Singapore Management University, 1974)
An HD-function (harmonic and Dirichlet-finite) ω on a Riemann surface R is called HD-minimal if $ω > 0$ and every HD-function ω' with 0 ≤ ω' ≤ ω reduces to a constant multiple of ω. An HD∼-function is the limit of a decreasing sequence of positive HD-functions and HD∼-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∼-minimal function but no HD-minimal functions?