The Dirichlet problem at infinity on Hadamard manifolds

Abstract

Let M be an n -dimensional Hadamard manifold, that is, a complete simply connected C ∞ Riemannian manifold with nonpositive sectional curvatures. Making use of geodesic rays, Eberlein and O’Neill [11] constructed a compactification = M S (∞) of M which gives a homeomorphism of ( M, S(∞ )) with the Euclidean pair ( B n , S n-1 ). In this paper we shall study the asymptotic Dirichlet problem for the Laplace-Beltrami operator, which is stated as follows: