## Duke Mathematical Journal

### Energy functionals and canonical Kähler metrics

#### Abstract

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 $E_k$ 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 $E_1$ 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 $E_1$ 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 $E_k$ are bounded below on the space of metrics with nonnegative Ricci curvature

#### Article information

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

Dates
First available in Project Euclid: 8 March 2007

https://projecteuclid.org/euclid.dmj/1173373453

Digital Object Identifier
doi:10.1215/S0012-7094-07-13715-3

Mathematical Reviews number (MathSciNet)
MR2309146

Zentralblatt MATH identifier
1116.32018

#### Citation

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

#### References

• T. Aubin, Équations du type Monge-Ampère sur les variétés kähleriennes compactes, Bull. Sci. Math. (2) 102 (1978), 63--95.
• S. Bando, The K-energy map, almost Einstein Kähler metrics and an inequality of the Miyaoka-Yau type, Tohoku Math. J. 39 (1987), 231--235.
• S. Bando and T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions'' in Algebraic Geometry (Sendai, Japan, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 11--40.
• X. X. Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 2000, no. 12, 607--623.
• —, On the lower bound of energy functional $E_1$, I: A stability theorem on the Kähler Ricci flow, J. Geom. Anal. 16 (2006), 23--38.
• —, private communication, 2005.
• X. X. Chen and G. Tian, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002), 487--544.
• —, Ricci flow on Kähler-Einstein manifolds, Duke Math. J. 131 (2006), 17--73.
• —, Geometry of Kähler metrics and foliations by holomorphic discs, preprint.
• P. Deligne, Le determinant de la cohomologie'' in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, 1987, 93--177.
• W. Y. Ding, Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann. 282 (1988), 463--471.
• W. Y. Ding and G. Tian, Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), 315--335.
• S. K. Donaldson, Scalar curvature and projective embeddings, I, J. Differential Geom. 59 (2001), 479--522.
• —, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289--349.
• —, Scalar curvature and projective embeddings, II, Q. J. Math. 56 (2005), 345--356.
• A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), 437--443.
• A. D. Hwang and G. Maschler, Central Kähler metrics with non-constant central curvature, Trans. Amer. Math. Soc. 355 (2003), 2183--2203.
• T. Mabuchi, K-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), 575--593.
• G. Maschler, Central Kähler metrics, Trans. Amer. Math. Soc. 355 (2003), 2161--2182.
• S. T. Paul and G. Tian, Analysis of geometric stability, Int. Math. Res. Not. 2004, no. 48, 2555--2591.
• D. H. Phong and J. Sturm, Stability, energy functionals, and Kähler-Einstein metrics, Comm. Anal. Geom. 11 (2003), 565--597.
• —, Scalar curvature, moment maps and the Deligne pairing, Amer. J. Math. 126 (2004), 693--712.
• —, The Futaki invariant and the Mabuchi energy of a complete intersection, Comm. Anal. Geom. 12 (2004), 321--343.
• —, On stability and the convergence of the Kähler-Ricci flow, J. Differential Geom. 72 (2006), 149--168.
• J. Ross and R. P. Thomas, A study of the Hilbert-Mumford criterion for the stability of projective varieties, preprint.
• Y. T. Siu, Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics, DMV Sem. 8, Birkhäuser, Basel, 1987.
• J. Song and B. Weinkove, On the convergence and singularities of the J-flow with applications to the Mabuchi energy, preprint.
• J. Sturm, personal communication, 2002.
• G. Tian, The K-energy on hypersurfaces and stability, Comm. Anal. Geom. 2 (1994), 239--265.
• —, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1--37.
• —, Canonical Metrics in Kähler Geometry, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2000.
• G. Tian and X. Zhu, A nonlinear inequality of Moser-Trudinger type, Calc. Var. Partial Differential Equations 10 (2000), 349--354.
• B. Weinkove, On the $J$-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom. 73 (2006), 351--358.
• S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), 339--411.
• —, Open problems in geometry'' in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, 1990), Proc. Sympos. Pure Math. 54, Part I, Amer. Math. Soc., Providence, 1993, 1--28.
• S. Zhang, Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), 77--105.