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

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

2011-09-06

arxiv.org/abs/1109.1263v1

Mathematics

Differential Geometry

34 pages, no figures

Scientific paper

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact K\"ahler 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-Amp\`ere equations of mean field (Liouville) type.

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