Energy functionals and canonical Kähler metrics

Abstract

Abstract. Yau 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, Donaldson 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 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. We show in fact 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.Mathematical Subject Classification: 32Q20 (Primary), 53C21 (Secondary)

The Moser-Trudinger inequality on Kähler-Einstein manifolds

“…Remark The results above also immediately imply analogous inequalities for the Mabuchi energy [13] by the argument in [22] and the E 1 functional [8] by the argument in [20].…”

Ricci iterations on Kähler classes

“…Since the K-energy is continuous this will be sufficient for the argument (Cf. [BM], [SW1,Section 3]). Now, for t ∈ [0, 1] one has…”

On energy functionals, Kähler–Einstein metrics, and the Moser–Trudinger–Onofri neighborhood

“…. , n. In this direction, an analogue of Theorem 1.1 for k = 1 was proved recently by Song and Weinkove [40]. The main purpose of the present article is to prove the following two statements.…”