Harmonic dimension and extremal length (Q1340418)

Let \(R\) be an open Riemann surface with a single boundary component and \(V\) an end of \(R\), that is a relatively noncompact subregion whose relative boundary \(\partial V\) consists of finitely many analytic Jordan curves. Let \({\mathcal P} (V)\) be the class of nonnegative harmonic functions on \(V\) with vanishing boundary values on \(\partial V\). M. Heins proved that if there is a sequence \((A_ n)\) of mutually disjoint annuli in \(V\) such that each \(A_{n+1}\) separates \(A_ n\) from the ideal boundary of \(V\) and such that the sum of the moduli of \(A_ n\) diverges, then \(\dim {\mathcal P} (V) = 1\). In this paper an example is given which shows that this condition is only sufficient and not necessary.