Measurable Riemann mapping theorem

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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 Alhfors 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.

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