Article

Energy functionals and canonical Kahler metrics

06/2005;
Source: arXiv

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.

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Keywords

canonical metric

conjecture

E_k functionals

Fano Kahler-Einstein manifold

Fano manifold

functional E_1

functionals E_k

G. Tian

geometric invariant theory

ideas

Kahler-Einstein metric

Mabuchi energy

Mabuchi energy functional

metrics

modulo holomorphic vector fields

nonnegative Ricci curvature

Tian

X.X. Chen

Yau conjectured