English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the method of finding conformal maps onto circle domains byapproximating with finitely connected subdomains. Every domain $D \subset\hat{C}$ admits exhaustions, i.e., increasing sequences of finitely connectedsubdomains $D_j$ whose union is $D$. By Koebe's theorem, each $D_j$ admits aconformal map $f_{D_j}$ from $D_j$ onto a circle domain $f_{D_j}(D_j)$.Assuming $f_{D_j} \to f$, our goal is to find out if $f(D)$ is also a circledomain.  We present a countably connected $D$ with an exhaustion $(D_j)$ so that$(f_{D_j})$ has a limit whose image is not a circle domain, and a domain$\Omega$ with an exhaustion $(\Omega_j)$ so that $(f_{\Omega_j})$ has a limitwhose image has uncountably many non-point complementary components.  On the other hand, we prove that every exhaustion $(D_j)$ of a countablyconnected $D$ admits a refinement so that the image of the corresponding limitmap is a circle domain. Our result extends the He-Schramm theorem on theuniformization of countably connected domains and provides a new proof.

Uniformization of planar domains by exhaustion