A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

Harmonic functions with certain boundedness properties on a given open Riemann surfaceRattain their maxima and minima on the harmonic boundaryΔBofR.The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12]. For the corresponding problem in higher dimensions we shall first show that the complement of ΔRwith respect to the Royden boundary ΓRof a Riemannian N-spaceRis harmonically negligible: given any non-empty compact subsetEof ΓR– ΔRthere exists an Evans superharmonic functionv,i.e., a positive continuous function onR* = R∪ΓR,superharmonic onR,withv= 0 on ΔR, v≡ ∞ onE,and with a finite Dirichlet integral overR.