Duke Mathematical Journal

Energy functionals and canonical Kähler metrics

Jian Song and Ben Weinkove

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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 Ek 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 E1 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 E1 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 Ek are bounded below on the space of metrics with nonnegative Ricci curvature

Article information

Duke Math. J., Volume 137, Number 1 (2007), 159-184.

First available in Project Euclid: 8 March 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]


Song, Jian; Weinkove, Ben. Energy functionals and canonical Kähler metrics. Duke Math. J. 137 (2007), no. 1, 159--184. doi:10.1215/S0012-7094-07-13715-3. https://projecteuclid.org/euclid.dmj/1173373453

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