2006/12/31 by Yanir A. Rubinstein · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:14J45 #msc:26B25 #msc:26D15 #msc:32Q20 #msc:32W20 #msc:53C25 #msc:58E11
paper · pdf · doi:10.1016/j.jfa.2007.10.009
published as J. Funct. Anal. 255, special issue dedicated to Paul Malliavin (2008), 2641-2660. · v2: 1. style and exposition changes made, 2. added Remark 4.5, 3. added several references. v3: added footnote on p. 8. v4: minor changes. To appear in Journal of Functional Analysis
arxiv created 2007/11/02 · openalex publication_date 2008/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We prove that the existence of a Kahler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kahler potentials on which this inequality holds on higher-dimensional Fano Kahler-Einstein manifolds.