Heins problem on harmonic dimensions

Abstract

The main assertion of this paper is that for an arbitrarily given parabolic open Riemann surface R there always exists a Heins surface W_R, i.e. a parabolic open Riemann surface with the single ideal boundary component, such that the harmonic dimension of W_R, i.e. the cardinal number of the set of minimal Martin boundary points of W_R, is identical with that of R. The result is then applied to give a simple and unified proof for the best theorem at present as an answer to the Heins problem to determine the set ∇ of harmonic dimensions of all Heins surfaces obtained by collecting contributions of many authors that ∇ contains the set N of all positive integers, the cardinal number $\aleph_0$ of countably infinite set, and the cardinal number $\aleph$ of continuum, i.e. $\nabla\supset\mathbf{N}\cup\{\aleph_0,\aleph\}$, so that $\nabla=[1,\aleph]$, the interval of cardinal numbers ξ with $1\leq\xi\leq\aleph$, when the continuum hypothesis is postulated.