Measurable Riemann mapping theorem

In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers ^[1] 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. ^[2]

Theorem

The theorem of Ahlfors and Bers states that if μ is a bounded measurable function on C with ${\displaystyle \|\mu \|_{\infty }<1}$, then there is a unique solution f of the Beltrami equation

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

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

1. Ahlfors, Lars; Bers, Lipman (September 1960). "Riemann's Mapping Theorem for Variable Metrics". The Annals of Mathematics. 72 (2): 385. doi:10.2307/1970141.
2. Morrey, Charles B. (January 1938). "On the Solutions of Quasi-Linear Elliptic Partial Differential Equations". Transactions of the American Mathematical Society. 43 (1): 126. doi:10.2307/1989904.

Further reading

• 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
• Zakeri, Saeed; Zeinalian, Mahmood (1996), "When ellipses look like circles: the measurable Riemann mapping theorem" (PDF), Nashr-e-Riazi, 8: 5–14