Article

# Energy functionals and canonical Kahler metrics

Duke Mathematical Journal (Impact Factor: 1.7). 06/2005; DOI: 10.1215/S0012-7094-07-13715-3

Source: arXiv

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**ABSTRACT:**An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow02/2008; - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kahler metrics, making contact with recent generalizations of the Calabi-Yau Theorem due to Dinew, Guedj-Zeriahi, and Kolodziej. As an application, we obtain a new analytic stability criterion for the existence of a Kahler-Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kahler-Ricci flow converges as soon as it converges in the metric sense.American Journal of Mathematics 02/2011; · 1.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we generalize Chen-Tian energy functionals to K\"ahler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of K\"ahler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic invariant. Comment: 18 pages06/2009;

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