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N-manifolds carrying bounded but no Dirichlet finite harmonic functions

Among the most remarkable results in the theory of harmonic functions on Riemann surfaces is the strictness of the inclusion relations OG < OHB < OHD , established by Ahlfors [1,2], Royden [2,4], and Tôki [8] two decades ago. Subsequently the strictness of the relations OG < OHP < OHB was shown and a somewhat simpler proof of OHB < OHD given by Sario [5] and Tôki [9]. Here OG is the class of parabolic surfaces, and OHP, OHB, OHD stand for the classes of surfaces which do not carry nonconstant harmonic functions which are positive, bounded, or Dirichlet finite, respectively.