Space of K\"ahler metrics III--On the lower bound of the Calabi energy and geodesic distance

Research paper by X. X. Chen

Indexed on: 09 Jun '06Published on: 09 Jun '06Published in: Mathematics - Differential Geometry


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