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.