Complex Monge-Ampère Operators in Analysis and Pseudo-Hermitian Manifolds∗

The paper is a short survey around the author’s recent works on topics related to complex Monge-Ampère equations and strictly pseudoconvex pseudo-Hermitian manifolds. 1. Invariant differential operators In complex analysis of one variable, the fact that the invariant property for Laplace operator under holomorphic change of coordinates plays an important role. Namely, Let φ : D1 → D2 be a holomorphic mapping, and let u be a smooth function on D2. If v(z) = u ◦ φ(z) for z ∈ D1, then (1.1) ∆v(z) = (∆u) ◦ φ(z)|φ′(z)|2 For example, this property couple with the existence, uniqueness and regularity theory of the Dirichlet problem of Laplacian was used beautifully by Painlevé [50] to prove the smooth extension for a proper holomorphic map between two smoothly bounded domains in the complex plane. However, Laplace operator is no longer invariant under holomorphical change of variables in C when n > 1. Instead, the complex Monge-Ampère operator is invariant under holomorphic change of coordinates. Let u be a real-valued function on a domain D in C, we let H(u)(z) denote the complex Hessian matrix [ ∂2u(z) ∂zi∂zj ] , which is an n× Hermitian matrix at each z ∈ D. Then the complex Monge-Ampère operator is defined as follows: (1.2) M [u](z) = detH(u)(z) = det [∂2u(z) ∂zi∂zj ] . ∗Primary subject 32V05, 32V20. Secondary subject 53C56.