On the continuity of half-plane capacity with respect to Carath\'eodory convergence

TL;DR

This paper proves the continuity of half-plane capacity with respect to Carathéodory convergence for unbounded hulls under certain conditions, extending to finitely connected domains.

Contribution

It establishes the continuity of half-plane capacity for unbounded hulls and extends the result to finitely connected domains under specific assumptions.

Findings

Half-plane capacity is continuous for unbounded hulls within a fixed hull.

Continuity extends to finitely connected domains under certain conditions.

Provides a framework for analyzing boundary hulls in complex analysis.

Abstract

We study the continuity of half-plane capacity as a function of boundary hulls with respect to the Carath\'eodory convergence. In particular, our interest lies in the case that hulls are unbounded. Under the assumption that every hull is contained in a fixed hull with finite imaginary part and finite half-plane capacity, we show that the half-plane capacity is indeed continuous. We also discuss the extension of this result to the case that the underlying domain is finitely connected.