ON THE CALABI ENERGY OF EXTREMAL KAHLER-METRICS

Fix a positive (1,1)-class on a compact Kahlerian manifold. Given a Kahler form representing this class, define its Calabi energy to be the L(2)-norm of its scalar curvature. This note proves that a critical metric for the Calabi energy, if any, is a global minimum among representatives of the chosen class, and that the critical value is determined a priori by the Kahler class. This answers affirmatively two questions of Calabi ([2, p. 99]).