15 March 2007 Energy functionals and canonical Kähler metrics
Jian Song, Ben Weinkove
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Duke Math. J. 137(1): 159-184 (15 March 2007). DOI: 10.1215/S0012-7094-07-13715-3


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


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Jian Song. Ben Weinkove. "Energy functionals and canonical Kähler metrics." Duke Math. J. 137 (1) 159 - 184, 15 March 2007. https://doi.org/10.1215/S0012-7094-07-13715-3


Published: 15 March 2007
First available in Project Euclid: 8 March 2007

zbMATH: 1116.32018
MathSciNet: MR2309146
Digital Object Identifier: 10.1215/S0012-7094-07-13715-3

Primary: 32Q20
Secondary: 53C21

Rights: Copyright © 2007 Duke University Press


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Vol.137 • No. 1 • 15 March 2007
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