Bounded harmonic functions on Riemannian manifolds of nonpositive curvature (Q431234)

Certain general conditions (I--IV) are imposed on a complete simply-connected Riemannian manifold \(M\) of nonpositive curvature, which guarantee that \(M\) supports nontrivial bounded harmonic functions. \newline Very roughly speaking the conditions (I--IV) are as follows. The sectional curvature of \(M\) is pinched between a negative constant and zero, the Ricci curvature of \(M\) is less than a negative constant. Removing a certain point \(O\in M\), there exists a topological fibration \(M \setminus \{ O\} = \mathcal{A} \times \mathcal {B}\), such that \(\mathcal{N}=\mathcal{A} \times [0, \infty) \) is a complete submanifold of \(M\) and \(O = (a^O, 0)\) for some \(a^O \in \mathcal{A}\). Some geometrical properties of the projection of \(M \setminus \{O\}\) to \(\mathcal{N}\) (i.e. suitable relations between the angle and its segments, for geodesic triangles) and some pinching on the sectional curvature of \(\mathcal{N}\) are required. \newline The main result provides new families of bounded harmonic functions on \(M\). In particular, it includes the Cartan--Hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains \(\Re_I(n,n)\) and \(\Re_{II}(n)\;(n\geqslant 2)\) as special cases.