Boundary Behavior of Bisectional Curvatures for Weighted Bergman Metrics

Abstract: This paper investigates the asymptotic boundary behavior of the holomorphic bisectional curvature for weighted Bergman metrics. By characterizing extremal functions via $L^2$-orthogonal projections, we establish an explicit formula for the weighted bisectional curvature. Utilizing the squeezing function, we then obtain quantitative upper and lower bounds for the curvature on bounded pseudoconvex domains. Furthermore, we prove that at strongly pseudoconvex boundary points, the bisectional curvature asymptotically coincides with that of the unit ball. As an application, these results provide a streamlined and unified proof for the known asymptotic behavior of the bisectional curvature of the Kähler-Einstein metric.