Invariant metrics on the Complex ellipsoid

TL;DR

This paper explores geometric convex domains where various invariant metrics are uniformly equivalent but not proportional, providing a full curvature tensor description in 2D and characterizing when these metrics are proportional.

Contribution

It introduces a class of convex domains with equivalent invariant metrics that are not proportional and characterizes the conditions for metric proportionality in 2D.

Findings

Invariant metrics are uniformly equivalent but not proportional on certain convex domains.

Full curvature tensor description for Bergman metric at boundary points in 2D.

Metrics are proportional only on the Poincaré disk.

Abstract

We provide a class of geometric convex domains on which the Carath\'eodory-Reiffen metric, the Bergman metric, the complete K\"ahler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Poincar\'e-disk.