In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and solutions of the Beltrami equation. The result was prefigured by earlier results of Charles Morrey from 1938 on quasi-linear elliptic partial differential equations.
Theorem The theorem of Ahlfors and Bers states that if μ is a bounded measurable function on C with ‖ μ ‖ ∞ < 1 {\displaystyle \|\mu \|_{\infty }<1} , then there is a unique solution f of the Beltrami equation
∂ z ¯ f ( z ) = μ ( z ) ∂ z f ( z ) {\displaystyle \partial _{\overline {z}}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 D. Their proof used the Beurling transform, a singular integral operator.
References
Further reading Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven (2009), Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton mathematical series, vol. 48, Princeton University Press, pp. 161–172, ISBN 978-0-691-13777-3 Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-97942-5 Zakeri, Saeed; Zeinalian, Mahmood (1996), "When ellipses look like circles: the measurable Riemann mapping theorem" (PDF), Nashr-e-Riazi, 8: 5–14