K\"ahler-Einstein fillings

TL;DR

This paper constructs a specific K"ahler-Einstein metric on strongly pseudoconvex domains in complex space, solving a complex Monge-Ampère equation with boundary conditions and establishing uniqueness under certain conditions.

Contribution

It introduces a method to find K"ahler-Einstein metrics with prescribed boundary behavior on pseudoconvex domains, advancing understanding of complex Monge-Ampère equations in this context.

Findings

Existence of K"ahler-Einstein metrics with positive Einstein constant on strongly pseudoconvex domains.

Solution of a complex Monge-Ampère equation with Dirichlet boundary conditions.

Uniqueness of the constructed metric under additional assumptions.

Abstract

We show that on an open bounded smooth strongly pseudoconvex subset of $\CC^{n}$, there exists a K\"ahler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Amp\`ere equation with Dirichlet boundary condition. We also prove uniqueness under some more assumptions on the open set.