Invariant Metrics on the Complex Ellipsoid

We provide a class of geometric convex domains on which the Carathéodory–Reiffen metric, the Bergman metric, the complete Kähler–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 Euclidean ball.