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 [math]\displaystyle{ \|\mu\|_\infty \lt 1 }[/math], then there is a unique solution f of the Beltrami equation
- [math]\displaystyle{ \partial_{\overline{z}} f(z) = \mu(z) \partial_z f(z) }[/math]
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
- 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, 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, https://archive.org/details/complexdynamics0000carl
- 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
- Zakeri, Saeed; Zeinalian, Mahmood (1996), "When ellipses look like circles: the measurable Riemann mapping theorem", Nashr-e-Riazi 8: 5–14, http://www.math.qc.edu/~zakeri/papers/ahl-bers.pdf
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