  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.
