• Source: 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



‖
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


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

Kata Kunci Pencarian:

• Riemann mapping theorem
• Measurable Riemann mapping theorem
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• List of things named after Bernhard Riemann
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• List of theorems
• Lars Ahlfors
• Quasiconformal mapping
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• Measurable function