Quasiconformal Mapping - Measurable Riemann Mapping Theorem

Measurable Riemann Mapping Theorem

Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable Riemann mapping theorem, proved by Morrey (1938). The theorem generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that D is a simply connected domain in C that is not equal to C, and suppose that is Lebesgue measurable and satisfies . Then there is a quasiconformal homeomorphism ƒ from D to the unit disk which is in the Sobolev space W1,2(D) and satisfies the corresponding Beltrami equation (1) in the distributional sense. As with Riemann's mapping theorem, this ƒ is unique up to 3 real parameters.