On the exponent of convergence of negatively curved manifolds without Green’s function

In this paper we prove that for every complete n-dimensional Riemannian manifold without Green’s function and with its sectional curvatures satisfying K ≤−1, the exponent of convergence is greater than or equal to n − 1. Furthermore, we show that this inequality is sharp. This result is well known for manifolds with constant sectional curvatures K = −1.