Abstrakt
It is shown that, for every noncompact parabolic Riemannian manifold X and every nonpolar compact K in X, there exists a positive harmonic function on X \ K which tends to infinity at infinity. (This is trivial for R, easy for R^2, and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space X, where constants are the only positive superharmonic functions and, for every nonpolar compact set K, there is a symmetric (positive) Green function for X \ K. This includes the case of parabolic Riemannian manifolds.
Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph.