Geometry and topology of the space of plurisubharmonic functions (Q1725885)

Let $\Omega$ be a strongly pseudoconvex domain. Define the Mabuchi space of strongly plurisubharmonic functions in $\Omega$ as \[ \mathcal{H}:=\{\varphi\in C^{\infty}(\bar{\Omega}, \mathbb{R})|dd^c\varphi\ge 0 ~\text{in}~ \bar{\Omega}~, \varphi=0 ~\text{on} ~\partial\Omega\}. \] In his first theorem, the author shows that the Mabuchi space equipped with the Levi-Civita connection is a locally symmetric space. \par The second main result in the paper under review is to establish regularity properties of geodesics in the ball by adapting the celebrated result of \textit{E. Bedford} and \textit{B. A. Taylor} [Invent. Math. 37, 1--44 (1976; Zbl 0315.31007)]. As an application, the author studies the existence of local Kähler-Einstein metrics in bounded smooth strongly pseudoconvex circled domains and proves that the existence of a solution to a certain family of Dirichlet problems is equivalent to the coercivity of the Ding functional.