Regularity of the Dirichlet problem for the complex Monge-Ampère equation

Abstract

Regularity up to the boundary of the solutions of a boundary value problem for a complex Monge-Ampere equation on perturbations of an annulus in Cn is proven. The result can be applied to the classification of such domains.During the past few years, several applications of the MongeAmpere equation, det (uj-p) = We prove the following result for general small perturbations of the ball:THEOREM. Given any k > 0 and 0 < R < 1, there exists an e > 0 such that for every bounded domain Q C Cn with CO' boundary which differs from the "annulus" Qo\Qi = [R < z < 1] by less than e in the C45 topology, the solution to Eq. 2 is of class Ck up to the boundary.In particular, the result of Bedford and Burns (ref. In the particular case uo = (log R)-' log Iz 1, the foliation is by complex planes through the origin and the operator is the laplacian on each leaf multiplied by a nonvanishing C-o function.(ii) For a small perturbation v of such a u, a two-dimensional real distribution D is defined by the eigenvectors corresponding to the minimum eigenvalues of the hessian matrix of v. It is not integrable in general, but a foliation F can be found with a tangent distribution close to the previous one. On each leaf we have a good substitute for polar coordinates (p, 0).The derivative P' can be perturbed to a second-order linear operator P' acting on each leaf of F separately, such that I I(Ps -PV)jl o1 < C (IIPVI 13-j + I IPVI II I IVI 15-j)

Complex Legendre duality

“…The applicability of this may be somewhat limited by the absence of positive existence results for geodesics, but if we change the set up slightly and consider functions ψ defined for t in a disk instead of a strip, there is at least one setting in which our theorem applies. Considering boundary data s → ψ s on the unit circle that happen to extend to a smooth solution of the HCMA, then the same thing holds for sufficiently small perturbations of the data [8], see also [16]. Taking the given boundary data to be identically equal to 0, for which trivially an extension exists, we see that any boundary data that are sufficiently small can be extended to a solution of the HCMA, ψ t with t in the disk ∆.…”

An Obstacle for Higher Regularity of Geodesics in the Space of Kähler Potentials

“…We denote 16) and consider them as elements of End(P 2n E ). To simplify (2.15), a standard idea is to find a family of automorphisms of P 2n…”

“…ds 2 (γ , γ ) 16 , (A. 16) the fact that a b (1+2a+2b) 2 is a constant 4 is equivalent to the fact that along the geodesic γ, the kinetic energy ds 2 (γ , γ ) is a constant.…”

Stability of envelopes of holomorphy and the degenerate Monge-Amp�re equation

“…A less technical, and more special, version of Theorem 5 may be stated as follows. Another approach to this problem is given in [14] and [15] by Moriy6n, who / O2U \ works more directly with the nonlinear operator det/~/\ zi02 fl and applies it to problem (1.8) in the case where q~" and qJ'~ are constant. Let us observe that Theorems 1 and 2 also give examples of polynomial hulls with piecewise smooth boundaries.…”