Simple proofs of uniformization theorems
| dc.creator | Glutsyuk, Alexey | |
| dc.date | 2005-10-04 | |
| dc.date.accessioned | 2026-07-07T06:20:52Z | |
| dc.date.available | 2026-07-07T06:20:52Z | |
| dc.description | 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. | |
| dc.description | 19 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/math/0510071 | |
| dc.identifier | http://arxiv.org/abs/math/0510071 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/95433 | |
| dc.subject | Complex Variables | |
| dc.subject | 32Q30; 32Q60 | |
| dc.title | Simple proofs of uniformization theorems | |
| dc.type | text |