Equivalence of invariant metrics via Bergman kernel on complete noncompact Kähler manifolds (Q6583560)

In this paper, various invariant metrics are compared to each other based on curvature assumptions. There are pre-existing results of this nature but they seem to rely on negative curvature pinching. For a natural class of examples (the pseudconvex domains \(E_{p,\lambda}\) as in the notation of the paper), these pre-existing results are not applicable. The main theorem (Theorem A) states that for a noncompact Kähler manifold with a complete Bergman metric of bounded curvature, if the Bergman kernel is comparable to the volume form, there is a negatively curved Kähler-Einstein metric that is comparable to the given one, and under the hypothesis of embeddability into the unit ball (and negative holomorphic sectional curvature outside a compact set), the Kobayashi-Royden metric is comparable to the base metric. The proofs use Shi's derivative estimates for the Kähler-Ricci flow as well as producing a negatively curved metric using the hyperbolic metric on the ball. In the last section, some sort of a lower bound is provided on an integral involving the Caratheodory-Reiffen metric.