  In this paper, we first prove a folklore conjecture on a greatest lower bound
of the Calabi energy in all K\"ahler manifold. Similar result in algebriac
setting was obtained by S. K. Donaldson. Secondly, we give an upper/lower bound
estimate of the K energy in terms of the geodesic distance and the Calabi
energy. This is used to prove a theorem on convergence of K\"ahler metrics in
holomorphic coordinates, with uniform bound on the Ricci curvature and the
diameter. Thirdly, we set up a framework for the existence of geodesic rays
when an asymptotic direction is given. I
