Energy functionals and canonical Kähler metrics

Abstract

Yau [Y2] has conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian [T1], [T2], Donaldson [Do1], [Do2], and others. The Mabuchi energy functional plays a central role in these ideas. We study the $E_k$ functionals introduced by X. X. Chen and G. Tian [CT1] which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kähler-Einstein metric, then the functional $E_1$ is bounded from below and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen [C2]. In fact, we show that $E_1$ is proper if and only if there exists a Kähler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kähler-Einstein manifold, all of the functionals $E_k$ are bounded below on the space of metrics with nonnegative Ricci curvature