Bergman metric on a Stein manifold with nonpositive constant holomorphic sectional curvature

TL;DR

This paper characterizes Stein manifolds with non-positive constant holomorphic sectional curvature, showing they are biholomorphic to the unit ball minus pluripolar sets, and explores the properties of their Bergman metrics.

Contribution

It proves that Stein manifolds with well-defined Bergman metrics of non-positive constant curvature are biholomorphic to the unit ball minus pluripolar sets, extending understanding of their geometric structure.

Findings

Stein manifolds with such Bergman metrics are biholomorphic to the unit ball minus pluripolar sets.

A Stein manifold cannot have a flat Bergman metric.

The paper constructs bounded strictly plurisubharmonic functions using Hörmander's L2-estimates and curvature conditions.

Abstract

We prove that the Bergman space of a Stein manifold separates points whenever its Bergman metric is well defined and has non-positive constant holomorphic sectional curvature. This, combined with earlier proved results, shows that a Stein manifold cannot admit a well-defined flat Bergman metric, and that it has a well-defined Bergman metric with negative constant holomorphic sectional curvature if and only if it is biholomorphic to the unit ball of the same dimension possibly with a pluripolar set removed. The proof is based on the Hormander L2-estimate for d-bar equations; and the curvature condition together with Calabi's rigidity and extension theorems is used to construct the required bounded strictly plurisubharmonic functions.