# Geometry and Topology of the space of plurisubharmonic functions

## TL;DR

This paper introduces a new geometric space of strongly plurisubharmonic functions on pseudoconvex domains, analyzes its metric properties via Mabuchi geodesics, and explores applications to local Kähler-Einstein metrics.

## Contribution

It defines the Mabuchi space for strongly plurisubharmonic functions and investigates its metric and regularity properties, especially in the ball, with applications to Kähler-Einstein metrics.

## Key findings

- Established regularity of Mabuchi geodesics in the ball
- Analyzed metric properties of the Mabuchi space
- Studied existence of local Kähler-Einstein metrics

## 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.