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

{{citation|first1=Lars|last1=Ahlfors|first2=Lipman|last2=Bers|title=Riemann's mapping theorem for variable metrics|journal= Annals of Mathematics|volume= 72|year=1960|issue=2 |pages= 385–404|doi=10.2307/1970141|jstor=1970141 }}
{{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966}}
{{citation|title=Elliptic partial differential equations and quasiconformal mappings in the plane|volume= 48|series= Princeton mathematical series|

isbn=978-0-691-13777-3|pages=161–172}}

{{citation|last1=Carleson|first1=L.|last2=Gamelin|first2=T. D. W.|title=Complex dynamics|series=Universitext: Tracts in Mathematics|publisher=Springer-Verlag|year=1993|isbn=0-387-97942-5|url-access=registration|url=https://archive.org/details/complexdynamics0000carl}}
{{citation

| title = On the solutions of quasi-linear elliptic partial differential equations

| first = Charles B. Jr.

| last = Morrey

| authorlink = Charles B. Morrey, Jr.

| journal = Transactions of the American Mathematical Society

| volume = 43

| year = 1938

| pages = 126–166

| doi = 10.2307/1989904

| issue = 1

| jstor = 1989904

| jfm = 62.0565.02

| mr = 1501936

| zbl = 0018.40501

| doi-access = free

}}

{{citation|title=When ellipses look like circles: the measurable Riemann mapping theorem|first1=Saeed|last1=Zakeri|first2=Mahmood|last2=Zeinalian|url=http://www.math.qc.edu/~zakeri/papers/ahl-bers.pdf|journal=Nashr-e-Riazi|volume= 8 |year=1996|pages=5–14}}

Category:Theorems in complex analysis

Category:Bernhard Riemann