Talk:Quasiconformal mapping
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A question
[edit]Whether addition of a K_1 quasiconformal map with a K_2 quasiconformal map is K_1+K_2 quasiconformal in the combined domain. (Unknown contributor)
Answer: Quasiconformal maps (qc maps) do no behave well under addition (f+g). A qc map must be a homeomorphism. But in some cases the sum of two qc maps is not a homeomorphism. For instance: f(x+iy)= x+i2y (a ℝ-linear map, 2-qc) and g(x+iy)=-x-2iy (a rotation, 1-qc); then (f+g)(x+iy)=iy is far from being injective, so not a homeomorphism. It flattens all ellipses so if you would want to give it a K it would have to be K=+∞. We can cook up wilder examples. Arnaud Chéritat (talk) 07:41, 12 July 2022 (UTC)
Yet: a small modification of your question is correct. It is true that the composition of a K1 and a K2 qc map is K1×K2 qc! Arnaud Chéritat (talk) 07:41, 12 July 2022 (UTC)