In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( $\neq \mathbb{R}^{2}$) can be mapped bijectively to the unit disk with a function that preserves angles between curves, ie is conformal. 


I have read the claim that conformal maps in higher dimensions are pretty boring but does anyone know a proof or even a intuitive argument that conformal maps in higher dimensions are trivial? 



Conformal maps in higher dimensions
I think you're looking for [Liouville's theorem](http://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)). 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.




---


EDIT : I'm updating this ancient answer to 
[link](http://lamington.wordpress.com/2013/10/28/liouville-illiouminated/) to a blog post by Danny Calegari which contains a sketch of a beautifully geometric argument for Liouville's theorem.


