A Selection Theorem for the Carath\'eodory Kernel Convergence of Pointed Domains

TL;DR

This paper establishes a selection theorem for domains in complex spaces, showing that certain sequences of pointed domains have subsequences converging to a kernel, aiding the understanding of holomorphic map families.

Contribution

It introduces a new selection theorem for tamed sequences of pointed domains in complex spaces, extending classical results to non-compact settings.

Findings

Sequences of tamed pointed domains have convergent subsequences in the Carathéodory sense.

The theorem generalizes the Blaschke selection theorem to complex domains.

It enhances the analysis of normal families of holomorphic maps with varying domains.

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 Carath\'eodory. 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.