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 $\Vert \mu {\Vert }_{\mathrm{\infty }}<1$, then there is a unique solution f of the Beltrami equation

${\partial }_{\bar{z}}f(z)=\mu (z){\partial }_{z}f(z)$

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

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