Energy functionals and canonical Kahler metrics

Duke Mathematical Journal (Impact Factor: 1.58). 06/2005; 137(1). DOI: 10.1215/S0012-7094-07-13715-3
Source: arXiv


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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    • "These works are mostly based on the continuity method, the Ricci flow and the J-flow. To cite a few papers from a rapidly growing literature, we mention [16] [43] [52] [19] [53] [46] [69] [68] and references therein. Perhaps one of the main thrusts of the present article is that it is considerably more powerful to use the Finsler geometry of H and various of its subspaces to treat in a unified manner essentially all instances of the properness conjectures. "
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    ABSTRACT: Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.
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    • "There are also some results on the lower bound of E k . Following a question proposed by X. X. Chen [3], Song-Weinkove [15] showed that the existence of Kähler- Einstein metrics is equivalent to the properness of E 1 in the canonical class, and they also showed that E k are bounded from below under some additional curvature conditions. Recently, following suggestion of X. X. Chen, the author [9] found new relations between all these functionals and generalized Pali-Song-Weinkove's results. "
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    ABSTRACT: Using Perelman's results on Kähler-Ricci flow, we prove that the $K$-energy is bounded from below if and only if the $F$-functional is bounded from below in the canonical Kähler class.
    OSAKA JOURNAL OF MATHEMATICS 03/2008; 45(1). · 0.40 Impact Factor
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    • "Theorem 1.3 generalizes some of Song-Weinkove's results in [14] "
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    ABSTRACT: We give a new formula for the energy functionals E_k defined by Chen-Tian, and discuss the relations between these functionals. We also apply our formula to give a new proof of the fact that the holomorphic invariants corresponding to the E_k functionals are equal to the Futaki invariant.
    International Mathematics Research Notices 10/2006; 2007. DOI:10.1093/imrn/rnm033 · 1.10 Impact Factor
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