Simple proofs of uniformization theorems

Details Simple proofs of uniformization theorems Simple proofs of uniformization theorems

2005-10-04

arxiv.org/abs/math/0510071v1

Mathematics

Complex Variables

19 pages, 2 figures

Scientific paper

The measurable Riemann mapping theorem proved by Morrey and in some particular cases by Ahlfors, Lavrentiev and Vekua, says that any measurable almost complex structure on $\rd$ ($S^2$) with bounded dilatation is integrable: there is a quasiconformal homeomorphism of $\rd$ ($S^2$) onto $\cc$ ($\bc$) transforming the given almost complex structure to the standard one. We give an elementary proof of this theorem that is done as follows. Firstly we prove its double-periodic version: each $\ci$ almost complex structures on the two-torus can be transformed by a diffeomorphism to the standard complex structure on appropriate complex torus. The proof is based on the homotopy method for the Beltrami equation on $\td$ with parameter. (As a by-product, we present a simple proof of the Poincar\'e-K\"obe theorem saying that each simply-connected Riemann surface is conformally equivalent to either $\bar{\cc}$, or $\cc$, or the unit disc.) Afterwards the general case is treated by $\ci$ double-periodic approximation and simple normality arguments (involving Gr\"otzsch inequality) following the classical scheme.

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