Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale"

Abstract

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact Kähler manifold X. In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of X (but there are counterexamples to the precise value proposed by Aubin). In the different setting of pseudoconvex domains in complex space we also obtain a quasi-sharp version of the inequalities and relate it to Brezis-Merle type inequalities. The inequalities are shown to be sharp for S^{1}-invariant functions on the unit-ball. We give applications to existence and blow-up of solutions to complex Monge-Ampère equations of mean field (Liouville) type.

34 pages, no figures

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• The projective space has maximal volume among all toric Kähler-Einstein manifolds
• On the Moser-Trudinger inequality in complex space
• Geodesics in the space of $m$-subharmonic functions with bounded energy
• Moser-Trudinger inequality for the complex Monge-Ampere equation
• Symmetrization of plurisubharmonic and convex functions
• Symmetrization of plurisubharmonic functions on the Fano manifolds
• A volume stability theorem on toric manifolds with positive Ricci curvature
• Geometry and Topology of the space of plurisubharmonic functions