Quasi-conformal mappings were first introduction by L. Ahlfors in 1935, in connection with open Riemann surfaces and value distribution theory, and then studied by O. Teichm\"uller in 1938. In 1956 A. Beurling and Ahlfors proved an extension theorem which states that a quasi-symmetric map $\phi:{\bold R\/} \to {\bold R\/}$ can be extended to a quasiconformal map of the upper half plane. In 1964 Ahlfors had proven a similar theorem for quasi-symmetric mappings of ${\bold R\/}^2$. Then in 1979 P. Tukia and J. V\"ais\"al\"a a proved quasiconformal extension to dimensions $\geq 5$. The case of extension from 3 to 4 dimensions is still unknown.
In this paper we prove three main quasi-conformal extension results. First we prove the Tukia-V\"ais\"al\"a theorem, which states that a quasiconformal map of upper half space ${\bold R_+^{n+1} \to {\bold R\/}_+^{n+1}$ provided $n \geq 5$. The second result is a similar extension theorem for the strongly uniform domains introduced by Heinonen and Yang 1993. Finally, we solve a problem posed Pansu in 1989 regarding the existence of a quasiconformal extension theorem for complex hyperbolic spaces. We can in fact prove this result for manifolds of pinched negative curvature.
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