Energy functionals and canonical Kahler metrics

TL;DR

This paper investigates the properties of generalized energy functionals on Fano manifolds, establishing a new criterion for the existence of Kahler-Einstein metrics linked to the boundedness and properness of the E_1 functional.

Contribution

It proves that the E_1 functional is proper if and only if a Fano manifold admits a Kahler-Einstein metric, providing a novel analytic criterion related to stability.

Findings

E_1 is bounded below on Fano manifolds with Kahler-Einstein metrics.

E_1 is proper if and only if a Kahler-Einstein metric exists.

All E_k functionals are bounded below on Fano Kahler-Einstein manifolds with nonnegative Ricci curvature.

Abstract

Yau conjectured that a Fano manifold admits a Kahler-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 Kahler-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 Kahler-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 Kahler-Einstein manifold all of the functionals E_k are bounded below on the space of metrics with nonnegative Ricci curvature.