In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by

Lars Ahlfors Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Background Ahlfors was born in Helsinki, Finland. His mother, Si ...

and

Lipman Bers Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also k ...

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...

and

geometric function theory Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem. Topics in geometric function theory The following are some of the most important topics in ge ...

. Contrary to its name, it is not a direct generalization of the

Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphic ma ...

, but instead a result concerning

quasiconformal mapping In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : '' ...

s and solutions of the

Beltrami equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation : = \mu . for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L'' ...

. The result was prefigured by earlier results of Charles Morrey from 1938 on quasi-linear elliptic partial differential equations. The theorem of Ahlfors and Bers states that if μ is a bounded measurable function on C with

\, \mu\, _\infty < 1

, then there is a unique solution ''f'' of the Beltrami equation :

\partial_ f(z) = \mu(z) \partial_z f(z)

for which ''f'' is a quasiconformal homeomorphism of C fixing the points 0, 1 and ∞. A similar result is true with C replaced by the

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...

D. Their proof used the Beurling transform, a singular integral operator.

References

* * * * * * Theorems in complex analysis Bernhard Riemann {{mathanalysis-stub