Huang–Li conjecture on Bergman metrics of positive constant curvature

Determine whether a pseudoconvex domain in complex Euclidean space whose Bergman space is base-point free and separates holomorphic directions can have a Bergman metric of positive constant holomorphic sectional curvature; equivalently, prove or refute the Huang–Li conjecture that such a domain cannot have this type of Bergman metric.

Background

The paper discusses a conjecture of Huang and Li concerning the interaction between the dimension and geometric properties of Bergman spaces. Specifically, Huang and Li conjectured that a pseudoconvex domain in complex Euclidean space with a base-point-free Bergman space that separates holomorphic directions cannot possess a Bergman metric of positive constant holomorphic sectional curvature.

The paper disproves Wiegerinck’s conjecture by constructing, in every complex dimension at least two, a pseudoconvex domain whose Bergman space is one-dimensional. However, the authors explicitly state that this construction does not resolve the Huang–Li conjecture, which therefore remains an open problem.

References

Motivated by these connections, Huang and Li conjectured that a pseudoconvex domain in $n$ whose Bergman space is base-point free and separates holomorphic directions cannot have a Bergman metric of positive constant holomorphic sectional curvature. By Theorem 1.1 in , Wiegerinck's conjecture would imply this conjecture of Huang and Li. Huang and Li subsequently constructed Stein manifolds of complex dimension $n$, embedded in ${n+1}$, whose Bergman metrics are well-defined and have positive constant holomorphic sectional curvature . Their work has stimulated further studies of Bergman metrics with positive constant holomorphic sectional curvature .

\medskip

In this paper, we construct a pseudoconvex domain in $n$ such that the Bergman space is one dimensional, for any $n \geq 2$. This provides counter-examples to Wiegerinck conjecture in every dimension. Nevertheless, the above deep conjecture of Huang-Li remains unsolved.

— Pseudoconvex domains in complex Euclidean space with one-dimensional Bergman spaces  (2610.21749 - Yuan, 15 Oct 2026) in Section 1, Introduction