Geometry and Topology of the space of plurisubharmonic functions

Abstract

Let $\Omega$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $\Omega$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local K\"ahler-Einstein metrics.