Energy functionals and canonical Kähler metrics

Jian Song, Ben Weinkove

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

You [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.

Original languageEnglish (US)
Pages (from-to)159-184
Number of pages26
JournalDuke Mathematical Journal
Volume137
Issue number1
DOIs
StatePublished - Mar 15 2007

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Fano Manifolds
Einstein Metrics
Metric
Energy
Geometric Invariant Theory
Holomorphic Vector Field
If and only if
Einstein Manifold
Nonnegative Curvature
Ricci Curvature
Energy Functional
Modulo
Generalise

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Song, Jian ; Weinkove, Ben. / Energy functionals and canonical Kähler metrics. In: Duke Mathematical Journal. 2007 ; Vol. 137, No. 1. pp. 159-184.
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Energy functionals and canonical Kähler metrics. / Song, Jian; Weinkove, Ben.

In: Duke Mathematical Journal, Vol. 137, No. 1, 15.03.2007, p. 159-184.

Research output: Contribution to journalArticle

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