Bounded harmonic functions on Riemannian manifolds of nonpositive curvature

Abstract

Certain general conditions are put forth on a complete simply-connected Riemannian manifold of nonpositive curvature which guarantee that they support nontrivial bounded harmonic functions. This result includes the Cartan–Hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains \({{\mathfrak{R}}_I(n,n)}\) and \({{\mathfrak{R}}_{II}(n)}\) (n ≥ 2) as special cases.