We give a complete and transparent proof of Koebe's General Uniformisation Theorem that every planar Riemann surface is biholomorphic to a domain in the Riemann sphere ℂ̂, by showing that a domain with analytic boundary and at least two boundary components on a planar Riemann surface is biholomorphic to a circular-slit annulus in ℂ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-1-7,
author = {Gollakota V. V. Hemasundar},
title = {Koebe's general uniformisation theorem for planar Riemann surfaces},
journal = {Annales Polonici Mathematici},
volume = {101},
year = {2011},
pages = {77-85},
zbl = {1222.30035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-1-7}
}
Gollakota V. V. Hemasundar. Koebe's general uniformisation theorem for planar Riemann surfaces. Annales Polonici Mathematici, Tome 101 (2011) pp. 77-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-1-7/