# Extension property in the tridisc

## TL;DR

This paper introduces a new concept of Carathéodory sets in the tridisc, explores special two-dimensional submanifolds, and investigates universal sets for the Carathéodory extremal problem, advancing understanding of extension properties in complex domains.

## Contribution

It proposes a refined notion of Carathéodory sets, identifies classes of submanifolds where Lempert's theorem applies, and studies the existence of finite universal sets in the tridisc.

## Key findings

- Certain two-dimensional submanifolds are Carathéodory and satisfy Lempert's theorem.
- New criteria for domains admitting finite universal sets are established.
- The introduced notion improves the characterization of extension sets in the tridisc.

## Abstract

Motivated by works on extension sets in standard domains we introduce a notion of the Carath\'eodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a class of two dimensional submanifolds in the tridisc which not only turns out to be Carath\'eodory but also provides examples of two dimensional domains for which the celebrated Lempert Theorem holds. Additionally, a recently introduced notion of universal sets for the Carath\'eodory extremal problem is studied and new results on domains admitting (no) finite universal sets are given.