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 μ -->
‖ ‖ -->
∞ ∞ -->
<
1
{\displaystyle \|\mu \|_{\infty }<1}
f of the Beltrami equation
∂ ∂ -->
z
¯ ¯ -->
f
(
z
)
=
μ 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 NostrandAstala, 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 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
