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's mapping theorem for variable metrics", Annals of Mathematics, 72 (2): 385–404, doi:10.2307/1970141, JSTOR 1970141
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
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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Cauchy–Riemann equations Generalized Riemann hypothesis Grand Riemann hypothesis Grothendieck–Hirzebruch–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Local zeta function Measurable Riemann mapping theorem Riemann (crater) Riemann Xi function Riemann curvature tensor Riemann hypothesis Riemann integral Riemann invariant Riemann mapping theorem Riemann form Riemann problem Riemann series theorem Riemann solver Riemann sphere Riemann sum Riemann surface Riemann zeta function Riemann's differential equation Riemann's minimal surface Riemannian circle Riemannian connection on a surface Riemannian geometry Riemann–Hilbert correspondence Riemann–Hilbert problems Riemann–Lebesgue lemma Riemann–Liouville integral Riemann–Roch theorem Riemann–Roch theorem for smooth manifolds Riemann–Siegel formula Riemann–Siegel theta function Riemann–Silberstein vector Riemann–Stieltjes integral Riemann–von Mangoldt formula
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