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Caratheodory sets in the tridisk

Lukasz Kosinski, John E. McCarthy

TL;DR

The paper classifies algebraic Carathéodory subsets of the tridisk $\\mathbb D^3$ by showing a dichotomy: such a set must be either a retract of $\\mathbb D^3$ or, after a biholomorphism, the exceptional two-variable-type set ${\\cal K}$ defined by $x+y+z=xy+yz+zx$ on $\\mathbb D^3$. The authors prove this via reduction to a two-dimensional parametrization and a detailed analysis of Carathéodory extremals, showing the latter case yields a rational map of degree two that forces a ${\\cal K}$-type structure; they also develop an extension-property result: under an isometric linear extension operator between $H^{\infty}$ spaces on a Cartan pair with convex, balanced, bounded $\\Omega$, the target set must be a retract under several common geometric conditions. Open questions address the extension property for the tridisk case of ${\\cal K}$, the interplay between extension-property and Carathéodory notions, and the possibility of Cartan pairs with Carathéodory-but-not-extension behavior. Overall, the work advances understanding of intrinsic Carathéodory geometry in several complex variables and the role of holomorphic extension in identifying retracts.

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.

Caratheodory sets in the tridisk

TL;DR

The paper classifies algebraic Carathéodory subsets of the tridisk by showing a dichotomy: such a set must be either a retract of or, after a biholomorphism, the exceptional two-variable-type set defined by on . The authors prove this via reduction to a two-dimensional parametrization and a detailed analysis of Carathéodory extremals, showing the latter case yields a rational map of degree two that forces a -type structure; they also develop an extension-property result: under an isometric linear extension operator between spaces on a Cartan pair with convex, balanced, bounded , the target set must be a retract under several common geometric conditions. Open questions address the extension property for the tridisk case of , the interplay between extension-property and Carathéodory notions, and the possibility of Cartan pairs with Carathéodory-but-not-extension behavior. Overall, the work advances understanding of intrinsic Carathéodory geometry in several complex variables and the role of holomorphic extension in identifying retracts.

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.
Paper Structure (6 sections, 16 theorems, 36 equations)

This paper contains 6 sections, 16 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. History
  3. Definitions and auxiliary results
  4. Carathéodory sets for $\mathbb D^3$
  5. Linear extension operator from $H^\infty({\mathcal{V}})$ to $H^\infty(\Omega)$
  6. Open Problems

Key Result

Theorem 1.2

Let ${\mathcal{V}}$ be an algebraic subset of $\mathbb D^3$. If ${\mathcal{V}}$ is a Carathéodory set for $\mathbb D^3$, then either ${\mathcal{V}}$ is a retract of $\mathbb D^3$, or, up to a biholomorphism of $\mathbb D^3$, ${\mathcal{V}} = {\cal K}$.

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 13 more