Existence of positive harmonic functions
Mitsuru Nakai · Proceedings of the American Mathematical Society · 1966
It is interesting to compare the theorem3 with the so-called domains due to Bader-Parreau [1] and Mori [4]: An open Riemann surface R does not belong to the class OHB (resp. OHD) if and only if there exist two disjoint distinguished subregions of R carrying nonconstant HB (resp. HD) functions with continuous boundary values zero on their relative boundaries. The theorem shows that the two domains criterion fails for the class OHP. Another consequence of the theorem is that the Martin compactification G* of any distinguished subregion G, considered as a Riemann surface, always contains an HP-minimal point other than those HP-minimal points identified with points in 9G, no matter whether REOG or not. For Martin's compactification and HP-minimal points see e.g. Constantinescu-Cornea [2]. The proof of the theorem will be given in ??2-4.