On the continuity of half-plane capacity with respect to Carath\'eodory convergence
Complex Variables
2022-11-09 v3 Probability
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.
Keywords
Cite
@article{arxiv.2106.12717,
title = {On the continuity of half-plane capacity with respect to Carath\'eodory convergence},
author = {Takuya Murayama},
journal= {arXiv preprint arXiv:2106.12717},
year = {2022}
}
Comments
19 pages, no figures. (v2)Two sections have been added: Section 2.2 on strict monotonicity and Appendix B on weak convergence of harmonic measures. Minor revisions have been made in other sections as well. (v3)Minor revision