Shenzhen University School of Mathematical Sciences  

Liyuan Scholars Colloquium Session 159

Title: Bergman metrics have constant holomorphic sectional curvatures

Speaker: Professor Song-Ying Li (University of California, Irvine)

Time: 16:00–17:00, January 7, 2026

Location: Room 1, Huixing Building, Yuehai Campus, Shenzhen University

Abstract:  I will talk about joint works with Xiaojun Huang from Rutgers University. We study domains in Cn or Stein manifolds M such that their Bergman metrics have constant holomorphic sectional curvature κ. A well-known theorem of Lu Qi-Keng states that any bounded domains Ω in C n whose Bergman metric is complete and has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to the unit ball Bn in C n . In practices, there are many domains whose Bergman metric are not complete but have constant holomorphic sectional curvatures. Recently, we generalize Lu’s theorem to unbounded domain whose Bergman metric may not be complete and prove the following theorem: Any Stein manifold of complex dimension n whose Bergman metric has non-positive holomorphic sectional curvatures if and only if it is biholomorphic to the unit ball in C n possible less a relatively closed pluripolor set. In the earlier paper, discussed Stein manifolds having positive constant holomorphic sectional curvature κ > 0. We first construct an interesting example of domain Ω ⊂ C2 so that its Bergman metric has holomorphic sectional curvature 2. Second, we prove that any complex manifold M of dimension n is Bergman separable and has positive constant holomorphic sectional curvature is biholomorphic to a domain in Pn with finite dimensional Bergman space A2 (M).

Speaker Profile:  Song-Ying Li, Professor at the University of California, Irvine (UC Irvine), is an internationally renowned expert in several complex variables. His research spans several complex variables, nonlinear partial differential equations, and harmonic analysis. He has published over 90 academic papers in internationally renowned journals including Amer. J. Math., Adv. Math., J. Funct. Anal., J. London Math. Soc., J. Differ. Geom., and Math. Ann.

All faculty and students are welcome!  

School of Mathematical Sciences  

January 5, 2026