Suppose that we have a homeomorphism of the real line onto itself, when and how can we extend this to a quasiconformal homeomorphism of the upper half plane to itself? In general we can consider any singly connected domain. First I will give a theorem by Beurling and Ahlfors that there exists a quasiconformal extension if and only if the boundary homeomorphism is a quasisymmetric mapping, that is when the boundary homeomorphism is the equivalent of a quasiconformal map in one dimension. The sufficiency part of the theorem is proved by an explicit construction of an extension. Later Douady and Earle proved, again by explicit construction, that we can in fact make a conformally natural extension. That is we can compose the boundary homeomorphism with two conformal automorphisms and then extend or first extend and then compose and we will in fact end up with the same map. We can then compare these two constructions and look at some applications.
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