Equivalence of Invariant metrics via Bergman kernel on complete noncompact Kähler manifolds

2021/09/29 by Cho, Gunhee, Lee, Kyu-Hwan
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.14473

Abstract

We study equivalence of invariant metrics on noncompact Kähler 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 Kähler manifold is required to obtain the equivalence of the Bergman metric and the complete Kähler--Einstein metric. To demonstrate the effectiveness of this method, we consider a two-parameter family of 3-dimensional bounded pseudoconvex domains E_p,λ=\(x,y,z)∈ ℂ³ ; (|x|^2p+|y|²)^1/λ+|z|²lt;1 \, p,λgt;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 Carathéodory--Reiffen metric on complete noncompact simply-connected Kähler manifolds with negative sectional curvature.

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