We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.
In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function on a given quasi-hyperconvex domain of a compact Kähler manifold , with well defined Monge-Ampère...
In this survey we present two Mittag-Leffler lemmas and several applications to topics as varied as the delta-equation, Fréchet algebras, inductive limits of Banach spaces and quasi-normable Fréchet spaces.
We study swept-out Monge-Ampère measures of plurisubharmonic functions and boundary values related to those measures.
Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....
We study estimates for the solution of the equation du=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solution to the equation.