Boundary Behavior of Bisectional Curvatures for Weighted Bergman Metrics

TL;DR

This paper studies the boundary behavior of holomorphic bisectional curvature in weighted Bergman metrics, providing explicit formulas, bounds, and asymptotic comparisons with the unit ball and Kähler-Einstein metrics.

Contribution

It introduces an explicit formula for weighted bisectional curvature and establishes its asymptotic boundary behavior using the squeezing function.

Findings

Explicit formula for weighted bisectional curvature.

Quantitative bounds on curvature in pseudoconvex domains.

Asymptotic coincidence with the unit ball's curvature at strongly pseudoconvex points.

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 K\"ahler-Einstein metric.