cv.complex variables - Conformal maps in higher dimensions

I think you're looking for Liouville's theorem. This theorem states that for $n >2$, if $V_1,V_2 subset mathbb{R}^n$ are open subsets and $f : V_1 rightarrow V_2$ is a smooth conformal map, then $f$ is the restriction of a higher-dimensional analogue of a Mobius transformation.

By the way, observe that there are no assumptions on the topology of the $V_i$ -- they don't have to be simply-connected, etc.

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EDIT : I'm updating this ancient answer to
link to a blog post by Danny Calegari which contains a sketch of a beautifully geometric argument for Liouville's theorem.