Abstract
The main assertion of this paper is that for an arbitrarily given parabolic open Riemann surface R there always exists a Heins surface WR, i.e. a parabolic open Riemann surface with the single ideal boundary component, such that the harmonic dimension of WR, i.e. the cardinal number of the set of minimal Martin boundary points of WR, 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 ℵ0 of countably infinite set, and the cardinal number ℵ of continuum, i.e. ∇⊃N∪{ℵ0, ℵ}, so that ∇=[1, ℵ], the interval of cardinal numbers ξ with 1≤ξ≤ℵ, when the continuum hypothesis is postulated.
