# K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two

## TL;DR

This paper establishes uniformization theorems for two-dimensional pseudoconvex domains of finite type using Kähler metrics, characterizing when such domains are biholomorphic to the unit ball and analyzing curvature properties.

## Contribution

It introduces new uniformization results for pseudoconvex domains of finite type in two dimensions using Kähler metrics with quasi-finite geometry, and characterizes the unit ball among Reinhardt domains.

## Key findings

- Pseudoconvex Reinhardt domain of finite type is the unit ball iff Bergman metric is a scalar multiple of Kobayashi or Carathéodory metric.
- Established a rigidity theorem for the holomorphic sectional curvature of the Bergman metric.
- Proved uniformization theorems via Kähler-Kobayashi and Kähler-Carathéodory metrics with quasi-finite geometry.

## Abstract

For a two dimensional bounded pseudoconvex domain of finite type, we prove uniformization theorems via K$\ddot{\operatorname{a}}$hler-Kobayashi metric or K$\ddot{\operatorname{a}}$hler-Carath$\acute{\operatorname{e}}$odory metric with quasi-finite geometry of order three. In particular, a pseudoconvex Reinhardt domain of finite type is the unit ball if and only if the Bergman metric is a scalar multiple of the Kobayashi metric or Carath$\acute{\operatorname{e}}$odory metric. Moreover, we establish a rigidity theorem concerning holomorphic sectional curvature of Bergman metric and Lu constant.