In this paper we prove that for every complete $n$-dimensional Riemannian manifold without Green's function and with its sectional curvatures satisfying $K \le -1$, 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 $K = -1$.
María V. Melián. José M. Rodríguez. Eva Tourís. "On the exponent of convergence of negatively curved manifolds without Green's function." Publ. Mat. 62 (1) 177 - 183, 2018. https://doi.org/10.5565/PUBLMAT6211809