ON SHARP LOWER BOUNDS FOR CALABI-TYPE FUNCTIONALS AND DESTABILIZING PROPERTIES OF GRADIENT FLOWS

Let X be a compact Kahler manifold with a given ample line bundle L. Donaldson proved an inequality between the Calabi energy of a Kahler metric in c(1)(L) and the negative of normalized Donaldson-Futaki invariants of test configurations of (X, L). He also conjectured that the bound is sharp. We prove a metric analogue of Donaldson\u27s conjecture; we show that if we enlarge the space of test configurations to the space of geodesic rays in epsilon(2) and replace the Donaldson-Futaki invariant by the radial Mabuchi K-energy M, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of M. On a Fano manifold, a similar sharp bound for the Ricci-Calabi energy is also derived