Le problème de Dirichlet pour les équationsde Monge–Ampère en métrique hermitienne

Complex Monge–Ampère equations and totally real submanifolds

“…Here we emphasize that C 1 and C 2 depend only on geometric quantities (torsion and curvature) of M and on χ as well as its covariant derivatives, but do not depend on inf ψ so the estimates (3.1) and (3.2) apply to the degenerate case (ψ ≥ 0); see Propositions 3.1 and 3.3 for details. For χ = ω these estimates were derived by Cherrier and Hanani [39], [40], [21], [22]. The estimate for ∆u is an extension of that of Yau [71].…”

“…An earlier version ( [31]) of this article was posted on the arXiv in June 2009. We learned afterwards of the work of Cherrier and Hanani [20], [39], [40], [21], [22]. We wish to thank Philippe Delanoë for bringing these beautiful papers to our attention.…”

On a class of fully nonlinear elliptic equations on Hermitian manifolds

“…The Dirichlet problem for the complex Monge-Ampère equation in C n was studied by Caffarelli, Kohn, Nirenberg and Spruck [3] on strongly pseudoconvex domains. Their result was extended to Hermitian manifolds by Cherrier and Hanani [8], [23], and by the first author [14] to arbitrary bounded domains in C n under the assumption of existence of a subsolution. See also the more recent papers [16], [35], and related work of Tosatti and Weinkove [29], [30] who completely extended the zero order estimate of Yau [34] on closed Kähler manifolds to the Herimatian case.…”

The Dirichlet problem for mixed Hessian equations on Hermitian manifolds

“…Then Tosatti-Weinkove [40,39] have solved the analogous problem for the equation on closed Hermitian manifolds. The corresponding Dirichlet problems on manifolds were studied by Cherrier-Hanani [9] and Guan-Li [16,17,18].…”