We seek to gain progress on the following long-standing conjectures in hyperbolic complex geometry: prove that a simply connected complete K ̈ahler manifold with negatively pinched sectional curvature is biholomorphic to a bounded domain and the Carath ́eodory-Reiffen metric does not vanish everywhere. As the next development of the important recent results of D. Wu and S.T. Yau in obtaining uniformly equivalence of the base K ̈ahler metric with the Bergman metric, the Kobayashi-Royden metric, and the complete Ka ̈hler-Einstein metric in the conjecture class but missing of the Carath ́eodory-Reiffen metric, we provide an integrated gradient estimate of the bounded holomorphic function which becomes a quantitative lower bound of the integrated Carath ́eodory-Reiffen metric. Also, without requiring the negatively pinched holomorphic sectional curvature condition of the Bergman metric, we establish the equivalence of the Bergman metric, the Kobayashi-Royden metric, and the complete Ka ̈hler-Einstein metric of negative scalar curvature under a bounded curvature condition of the Bergman metric on an n-dimensional complete noncompact Ka ̈hler manifold with some reasonable conditions which also imply non-vanishing Carath ́edoroy-Reiffen metric. This is a joint work with Kyu-Hwan Lee.