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

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.

💡 Research Summary

The paper investigates the relationship between the Bergman space (A^{2}(M)), the Bergman metric (\omega_{B}), and the holomorphic sectional curvature on a Stein manifold (M). The Bergman space consists of square‑integrable holomorphic (n)-forms; when it is non‑trivial and base‑point free, the Bergman kernel (K_{M}) is nowhere zero and defines a Kähler metric (\omega_{B}= \sqrt{-1},\partial\bar\partial\log k_{M}). The authors assume that (\omega_{B}) is well‑defined and that its holomorphic sectional curvature is a non‑positive constant (c\le 0).

The first main result (Theorem 1.1) shows that under these hypotheses the Bergman space separates points: for any distinct points (p_{1},p_{2}\in M) there exists a form (\varphi\in A^{2}(M)) with (\varphi(p_{1})=0) and (\varphi(p_{2})\neq0). The proof proceeds by constructing a bounded strictly plurisubharmonic function (\sigma) on (M) (possible on any Stein manifold) and then applying Hörmander’s (L^{2}) (\bar\partial)‑estimate to solve a (\bar\partial)-equation with a carefully chosen right‑hand side that forces the resulting holomorphic form to vanish at one point but not at the other. The curvature condition guarantees that the Bochner–Kodaira–Morrey–Kohn identity yields a positive lower bound for the weighted (L^{2}) norm, which is essential for the Hörmander estimate.

The second main result (Theorem 1.2) treats the case of strictly negative constant curvature. By combining Theorem 1.1 with earlier work of the authors (