Subharmonicity of the variations of K\"ahler-Einstein metrics on pseudoconvex domains

TL;DR

This paper investigates the subharmonicity of variations of K"ahler-Einstein metrics on bounded strongly pseudoconvex domains in complex dimension 2, extending previous results to lower dimensions and more general domains.

Contribution

It extends the study of subharmonicity of K"ahler-Einstein metrics to complex dimension 2 and discusses the behavior on general pseudoconvex domains and local triviality.

Findings

Established subharmonicity of metric variations in dimension 2

Extended results to general bounded pseudoconvex domains

Analyzed local triviality of domain families

Abstract

This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of K\"ahler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations K\"ahler-Einstein metrics on bounded strongly pseudoconvex domains of dimension $2$. In addition, we discuss the previous result with general bounded pseudoconvex domain and local triviality of a family of bounded strongly pseudoconvex domains.