Caratheodory sets in the tridisk

TL;DR

This paper characterizes all algebraic subsets of the tridisk that are Caratheodory sets, identifying the conditions under which the intrinsic and ambient Caratheodory metrics coincide, revealing a classification involving retracts and an exceptional set.

Contribution

It provides a complete characterization of Caratheodory sets in the tridisk, including a classification into retracts and a unique exceptional set.

Findings

Caratheodory sets are either retracts or isomorphic to an exceptional set

The intrinsic Caratheodory metric coincides with the ambient metric on these sets

Complete classification of algebraic subsets with this property in the tridisk

Abstract

We characterize all algebraic subsets of the tridisk that are Caratheodory sets, that is the intrinsic Caratheodory metric on the set equals the Caratheodory metric for the tridisk. We show that such sets are either retracts, or are isomorphic to one particular exceptional set.