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

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 $$E_{p,\lambda }=\{(x,y,z)\in {\mathbb{C}}^3;(\mathrm{\mid }x{\mathrm{\mid }}^{2p}+\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{1\mathrm{/}\lambda }+\mathrm{\mid }z{\mathrm{\mid }}^2<1\},p,\lambda >0.$$ 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.