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A Selection Theorem for the Carathéodory Kernel Convergence of Pointed Domains

Kang-Tae Kim, Thomas Pawlaschyk

TL;DR

The paper introduces a notion of Carathéodory kernel convergence for sequences of pointed domains in $\mathbb{C}^n$ and proves a selection theorem guaranteeing a subsequence converges to its kernel $Ker_{\hat p}\{(G_j,p_j)\}$. It then establishes a high-dimensional Carathéodory kernel theorem: when biholomorphic maps between tamed domain pairs converge, the limits between the kernels are biholomorphisms, with the subsequential limits preserving invertibility. The authors provide explicit kernel-computation tools, including monotone formulas, the pre-kernel construction, and a Psi-based framework for sublevel set kernels. Overall, the work furnishes a robust topological framework connecting kernel limits to normal-family phenomena and high-dimensional holomorphic mapping, enabling analysis of varying domains and their holomorphic images.

Abstract

We present a selection theorem for domains in $\mathbb{C}^n$, $n\ge 1$, which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carathéodory. Not only is this analogous to the well-known Blaschke selection theorem for compact convex sets, but it fits better in the study of normal families of holomorphic maps with varying domains and ranges.

A Selection Theorem for the Carathéodory Kernel Convergence of Pointed Domains

TL;DR

The paper introduces a notion of Carathéodory kernel convergence for sequences of pointed domains in and proves a selection theorem guaranteeing a subsequence converges to its kernel . It then establishes a high-dimensional Carathéodory kernel theorem: when biholomorphic maps between tamed domain pairs converge, the limits between the kernels are biholomorphisms, with the subsequential limits preserving invertibility. The authors provide explicit kernel-computation tools, including monotone formulas, the pre-kernel construction, and a Psi-based framework for sublevel set kernels. Overall, the work furnishes a robust topological framework connecting kernel limits to normal-family phenomena and high-dimensional holomorphic mapping, enabling analysis of varying domains and their holomorphic images.

Abstract

We present a selection theorem for domains in , , which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carathéodory. Not only is this analogous to the well-known Blaschke selection theorem for compact convex sets, but it fits better in the study of normal families of holomorphic maps with varying domains and ranges.
Paper Structure (6 sections, 12 theorems, 37 equations)

This paper contains 6 sections, 12 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The Carathéodory kernel convergence
  3. A selection theorem for the kernel convergence
  4. On Carathéodory's kernel theorem in all dimensions
  5. On computations of kernels
  6. Final remarks

Key Result

Proposition 2.7

A pointed domain $(\hat{G}, \hat{p})$ is the normal limit of the sequence $\{(G_j, p_j)\}_{j\geq1}$ of pointed domains tamed at $\hat{p}$ if and only if $(\hat{G}, \hat{p}) = \textup{Ker}_{\hat{p}} \{ (G_j, p_j)\}_{j\geq1}$.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Example 2.8
  • ...and 28 more