Invariant metrics on negatively pinched complete K\"ahler manifolds

We prove that a complete K\"ahler manifold with holomorphic curvature bounded between two negative constants admits a unique complete K\"ahler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background K\"ahler metric. Furthermore, all three metrics are shown to be uniformly equivalent to the Bergman metric, if the complete K\"ahler manifold is simply-connected, with the sectional curvature bounded between two negative constants. In particular, we confirm two conjectures of R. E. Greene and H. Wu posted in 1979.