K\"ahler-Einstein Bergman metrics on pseudoconvex domains of dimension two

TL;DR

This paper proves that two-dimensional pseudoconvex domains with a Kähler-Einstein Bergman metric are biholomorphic to the unit ball, answering a longstanding question of Yau for such domains.

Contribution

It establishes a characterization of certain pseudoconvex domains via their Bergman metrics, linking geometric properties to biholomorphic equivalence.

Findings

Domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

The proof uses asymptotics of Bergman kernel derivatives along tangent paths.

Answers an old question of Yau regarding domain classification.

Abstract

We prove that a two dimensional pseudoconvex domain of finite type with a K\"ahler-Einstein Bergman metric is biholomorphic to the unit ball. This answers an old question of Yau for such domains. The proof relies on asymptotics of derivatives of the Bergman kernel along critically tangent paths approaching the boundary, where the order of tangency equals the type of the boundary point being approached.