On energy functionals, Kahler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

Abstract

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.

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

References in corpus (4)

• Ricci flow on Kähler-Einstein manifolds
• Notes on GIT and symplectic reduction for bundles and varieties
• On the Critical Points of the E_k Functionals in Kahler Geometry
• The Moser-Trudinger inequality on Kahler-Einstein manifolds

Cited by in corpus (8)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
• The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Convergence of the Kähler-Ricci iteration
• Tian's properness conjectures: an introduction to Kahler geometry
• A derivation of the sharp Moser-Trudinger-Onofri inequalities from the fractional Sobolev inequalities