Unbounded Reinhardt domains with finite-dimensional Bergman spaces in $\C^n$

TL;DR

The work constructs unbounded complete Reinhardt domains $D(a)$ in $\mathbb{C}^n$ ($n\ge 2$) with nontrivial finite-dimensional Bergman spaces, extending 2D examples to arbitrary dimension. For certain ranges of $a$, the Bergman metric has constant holomorphic sectional curvature $2$ and the Bergman kernel reduces to a quadratic form after scaling, revealing a deep link to the Fubini-Study metric. Under these conditions, the holomorphic automorphism group of $D(a)$ is linear, with all automorphisms fixing the origin. Overall, the paper connects Bergman space dimensions, curvature properties, and automorphism structure, enriching the understanding of unbounded domains in several complex variables and generalizing known 2D phenomena to higher dimensions.

Abstract

In this paper, we construct unbounded domains in $\C^n$ ($n\geq 2$), whose Bergman spaces are nontrivial and finite-dimensional. We further show that the Bergman metrics on these domains have positive constant sectional curvature equal to $2$, and that their holomorphic automorphism groups consist only of linear mappings.

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Remark 2.2
• Theorem 3.1
• proof
• Theorem 4.1: IsK10Yam14KNY14
• Lemma 4.2: Yam17
• Theorem 4.3
• proof
• Theorem 4.4