A Measure on the Harmonic Boundary of a Riemann Surface

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Abstract
In the usual theory of harmonic functions on a plane domain, the fact that the boundary of the domain is realized relative to the complex plane plays an essential role and supplies many powerful tools, for instance, the solution of Dirichlet problem. But in the theory of harmonic functions on a general domain, i.e. on a Riemann surface, the main difficulty arises from the lack of the “visual” boundary of the surface. Needless to say, in general we cannot expect to get the “relative” boundary with respect to some other larger surface. In view of this, we need some “abstract” compactifications. It seems likely that we cannot expect to get the “universal” boundary which is appropriate for any harmonic functions since there exist many surfaces which do not admit some classes of harmonic functions as the classification theory shows. Hence we need many compactifications corresponding to what class of harmonic functions we are going to investigate.

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