Measurable Riemann mapping theorem

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.

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 _{{\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

Ahlfors, Lars; Bers, Lipman (1960), "Riemann mapping's theorem for variable metrics", Annals of Math., 72: 385–404
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, 48, Princeton University Press, pp. 161–172, ISBN 0-691-13777-3
Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-97942-5
Morrey, Charles B. Jr. (1938), "On the solutions of quasi-linear elliptic partial differential equations", Transactions of the American Mathematical Society, 43 (1): 126–166, doi:10.2307/1989904, JFM 62.0565.02, JSTOR 1989904, MR 1501936, Zbl 0018.40501
Zakeri, Saeed; Zeinalian, Mahmood (1996), "When ellipses look like circles: the measurable Riemann mapping theorem" (PDF), Nashr-e-Riazi, 8: 5–14

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