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

TL;DR

The paper investigates when the Bergman space $A^2(M)$ separates points on a Stein manifold $M$ under the Bergman metric with nonpositive constant holomorphic sectional curvature. It proves that such curvature implies point separation, and, in conjunction with prior results, that the curvature must be $c=-2/(n+1)$ and $M$ is biholomorphic to the unit ball ${f B}^n$ with a pluripolar set removed, i.e., $M\

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.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• Theorem 3.2
• proof : Proof of Proposition \ref{['1-3']}
• proof : Proof of Theorem \ref{['mainthm3']}
• Proposition 4.1
• Theorem 4.2
• proof : Proof of Proposition \ref{['thm2.1']}
• proof : Proof of Theorem \ref{['mainthm1']}