Measurable Riemann mapping theorem

In the mathematical theory of quasiconformal mappings in two dimensions, the measurable Riemann mapping theorem, proved by Morrey (1936, 1938), 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 $\mu:D\to \mathbf C$ is Lebesgue measurable and satisfies $\|\mu\|_\infty<1$. Then there is a quasiconformal homeomorphism ƒ from D to the unit disk which is in the Sobolev space W^1,2(D) and satisfies the corresponding Beltrami equation in the distributional sense. As with Riemann's mapping theorem, this ƒ is unique up to 3 real parameters.

References

• Morrey, Charles B. (1936), "On the solutions of quasi-linear elliptic partial differential equations.", Bulletin of the American Mathematical Society 42 (5): 316, doi:10.1090/S0002-9904-1936-06297-X, ISSN 0002-9904, JFM 62.0565.02 
• Morrey, Charles B. Jr. (1938), "On the Solutions of Quasi-Linear Elliptic Partial Differential Equations", Transactions of the American Mathematical Society (American Mathematical Society) 43 (1): 126–166, doi:10.2307/1989904, JSTOR 1989904, Zbl 0018.40501 .