Abstract
This chapter explores harmonic functions with finite growth on a manifold of asymptotically nonnegative curvatures. A complete connected noncompact Riemannian manifold M abounds harmonic functions so that M can be imbedded properly into some Euclidean space by them. If M is a complete connected noncompact Riemannian manifold such that the sectional curvature is bounded from below by c/r2 log r outside a compact set, where c is a positive constant and r is the distance to a fixed point of M, then M has no nonconstant harmonic functions of finite growth if M has only one end.
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