The Moser-Trudinger inequality on Kähler-Einstein manifolds

Abstract

ABSTRACT:We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kähler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.

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“…Song and Weinkove [2008] found a necessary and sufficient condition for the convergence of the J -flow in higher dimensions, which we now explain. Define…”

“…It also holds on all manifolds with c 1 (X ) = 0, even in the presence of holomorphic vector fields [Tian 2000]. In fact in each case, the function f can be taken to be linear [Tian 2000;Phong et al 2008]. Chen [2000] showed that on manifolds with c 1 (X ) < 0, or equivalently, with ample canonical bundle K X , the Mabuchi energy can be written as a sum of two terms: the first is the -functional with reference metric ω in [K X ], and the second is a term which is bounded below.…”

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“…The convergence on the level of Kähler metrics in the Fano case, that is, when K X is ample, was proved by Perelman (unpublished) and Tian and Zhu [2007]. The convergence on the level of weights then follows directly from Proposition 2.4 and the known coercivity of the functionals Ᏺ˙; the coercivity of Ᏺ C follows immediately from Jensen's inequality, while the coercivity of Ᏺ was shown in [Phong et al 2008], confirming a conjecture of Tian. The uniqueness in the difficult case of K X is due to Bando and Mabuchi (for a comparatively simple proof see [Berman et al 2009]).…”

“…In fact, in the present notation h t coincides (modulo signs) with the time derivative of t evolving according to the normalized Kähler-Ricci flow in the K X -setting. The second key ingredient is the fact that the existence of a Kähler-Einstein metric implies that Ᏺ C is proper (and conversely [Tian 2000;Phong et al 2008]). As is well-known there are, in general, obstructions to existence of Kähler-Einstein metrics in the K X -setting.…”