Simple proofs of uniformization theorems

Abstract

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 Poincaré-Köbe 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 Grötzsch inequality) following the classical scheme.