Comparing the Carath\'eodory Pseudo-Distance and the K\"ahler-Einstein Distance on Complete Reinhardt Domains

TL;DR

This paper demonstrates that on symmetric, complete Reinhardt domains in complex space, the Carathéodory pseudo-distance and the Kähler-Einstein geodesic distance do not coincide, highlighting differences in these intrinsic metrics.

Contribution

It provides a comparison between two fundamental complex geometric distances on a specific class of symmetric domains, showing they are not equivalent.

Findings

Carathéodory and Kähler-Einstein distances differ on certain domains

Symmetries of Reinhardt domains influence metric equivalences

Explicit distinction established between the two distances

Abstract

We show that on a certain class of bounded, complete Reinhardt domains in $\mathbb{C}^n$ that enjoy a lot of symmetries, the Carath\'eodory pseudo-distance and the geodesic distance of the complete K\"ahler-Einstein metric with Ricci curvature $-1$ are different.