Beurling-Ahlfors quasiconformal extension
There exists a quasiconformal mapping of the upper half plane to itself if and only if the boundary correspondence mapping is -quasisymmetric (http://planetmath.org/QuasisymmetricMapping). Furthermore there exists an extension of to a quasiconformal mapping of the upper half planes such that the maximal dilatation of the extension depends only on and not on .
That is, the extension is -quasiconformal (http://planetmath.org/QuasiconformalMapping) if and only if the boundary correspondence is -quasisymmetric (http://planetmath.org/QuasisymmetricMapping) where depends purely on .
Supposing that we have the mapping (where is the upper half plane), then the mapping , such that where , is the boundary correspondence of .
Intuitively is a function which “smoothes” out any kinks in the function as we get further and further away from the real line. It therefore intuitively follows that has the worst (highest) dilatation near the axis, which actually turns out to be true.
- 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
- 2 A. Beurling, L. V. Ahlfors. . Acta Math., 96:125-142, 1956.
- 3 J. Lebl. . . Also available at http://www.jirka.org/thesis.pdfhttp://www.jirka.org/thesis.pdf
|Title||Beurling-Ahlfors quasiconformal extension|
|Date of creation||2013-03-22 14:06:49|
|Last modified on||2013-03-22 14:06:49|
|Last modified by||jirka (4157)|