Heins problem on harmonic dimensions (Q2574395)

Let \(\Delta_1(R)\) denote the set of minimal points of the Martin boundary of an open Riemann surface \(R\). The \textit{harmonic dimension} of \(R\) is the cardinal number of \(\Delta_1(R)\). A parabolic open Riemann surface \(R\) is a \textit{Heins surface} if it has a single ideal boundary component (equivalently, for any compact subset \(K\) of \(R\), the set \(R\setminus K\) has only one relatively noncompact component). The \textit{Heins problem} is to determine the set \[ \nabla=\{\text{dim } R: R\;\text{ Heins surface}\}. \] The main result of the paper under review is that given a parabolic Riemann surface, there exists a Heins surface with the same harmonic dimension. From this it follows that \[ \mathbb N\cup \{\aleph_0, \aleph\}\subset \nabla\subset [1,\aleph]. \] This is the best known result on Heins problem; it has been proved by various authors during the last sixty years. Under the continuum hypothesis, \(\nabla=[1,\aleph]\) and Heins problem is solved; otherwise it is open. The paper also contains an introduction on the definition of Martin boundary for parabolic Riemann surfaces (where there is no Green function), on the definition of Heins surfaces, and on the history of the Heins problem.