Le problème de Dirichlet pour des équations de Monge–Ampère complexes modifiées

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].…”

“…The classical solvability of the Dirichlet problem was established by Caffarelli, Kohn, Nirenberg and Spruck [12] for strongly pseudoconvex domains in C n . Their results were extended to strongly pseudoconvex Hermitian manifolds by Cherrier and Hanani [21], [22] (for χ = 0, ω, −uω in (1.1)), and to general domains in C n by the first author [30] under the assumption of existence of a subsolution. This latter extension and its techniques have found useful applications in some important work; see, e.g., P.-F. Guan's proof [35], [36] of Chern-Levine-Nirenberg conjecture [19] and the papers of Chen [16], Blocki [10], and Phong and Sturm [56] on the Donaldson conjectures [23].…”

“…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.…”

The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary

“…When m = n we need not assume α is locally conformal Kähler. The Dirichlet problem for the Monge-Ampère equation on compact Hermitian manifolds with boundary has been studied extensively, in smooth category, in recent years by Cherrier-Hanani [15,16], Guan-Li [28] and Guan-Sun [29]. Our theorem generalises the result in [28] for continuous datum.…”

Regularity estimates of solutions to complex Monge–Ampère equations on Hermitian manifolds

“…In the eighties and nineties, some results regarding the Monge-Ampère equation in the Hermitian setting were obtained by Cherrier [3], [4] and Hanani [10]. For next few years there was no activity on the subject until very recently, when the results were rediscovered and generalized by Guan-Li [9].…”