On Point Separation of Bergman Space on Stein Manifolds with Constant Holomorphic Sectional Curvature

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. We construct examples of Stein manifolds whose Bergman metric is well defined and has positive constant holomorphic sectional curvature, while their Bergman spaces do not separate points. We also construct examples of Stein manifolds whose Bergman metric is well defined and has constant scalar curvature, which can be negative, zero, or positive, yet whose Bergman spaces do not separate points. Combined with previously established results in [HuLi1], this shows that a Stein manifold cannot admit a well-defined flat Bergman metric, and that it admits 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. Our proof is based on Hörmander's $L^2$ estimates for $\overline\partial$- equations; the curvature condition, together with Calabi's rigidity and extension theorems, is used to construct the required bounded strictly plurisubharmonic functions. The construction of Stein manifolds with positive constant holomorphic sectional curvature for their Bergman metric is based on classical hyperelliptic Riemann surface theory and its higher-dimensional generalizations.