Equivalence of Invariant metrics via Bergman kernel on complete noncompact K\"ahler manifolds

TL;DR

This paper establishes conditions under which invariant metrics on complete noncompact K"ahler manifolds are equivalent, using Bergman kernel estimates, and applies this to complex domains with challenging curvature properties.

Contribution

It introduces a new method based on Bergman kernel ratios to prove metric equivalence on noncompact K"ahler manifolds with bounded curvature.

Findings

Bergman metric and K"ahler--Einstein metric are equivalent under certain boundedness conditions.

The method applies to complex domains with non-well-defined boundary curvature limits.

Provides a lower bound estimate for the Carathéodory--Reiffen metric on negatively curved K"ahler manifolds.

Abstract

We study equivalence of invariant metrics on noncompact K\"ahler manifolds with a complete Bergman metric of bounded curvature. Especially only the boundedness of the ratio between Bergman kernel and the $n$-times wedge product of Bergman metric in any fundamental domain of such a K\"ahler manifold is required to obtain the equivalence of the Bergman metric and the complete K\"ahler--Einstein metric. To demonstrate the effectiveness of this method, we consider a two-parameter family of $3$-dimensional bounded pseudoconvex domains \[ E_{p,\lambda}=\{(x,y,z)\in \mathbb{C}^3 ; (|x|^{2p}+|y|^2)^{1/{\lambda}}+|z|^2<1 \},\qquad p,\lambda>0.\] For this family, boundary limits of the holomorphic sectional curvature of the Bergman metric are not well-defined, and hence previously known methods for comparison of invariant metrics do not work. Lastly, we provide an estimate of lower bound of the integrated Carath\'eodory--Reiffen metric on complete noncompact simply-connected K\"ahler manifolds with negative sectional curvature.