TITLE: 3D conformal mappings
QUESTION [8 upvotes]: Are there analogues to conformal mapping in 3 dimensions? 
I have a specific example I am trying to solve.. Laplace's equation in 3D with slightly complicated rectilinear boundaries. (Think of solving a harmonic function over a 3D boundary which is a cube but with a sub-cube "bitten" out of one corner.)
Laplace's equation is still valid under conformal transformations, so for example in 2D I could take a square domain with a subsquare bitten out of a corner, and apply an inverse tranformation like some of these and solve the equation in a simple square domain. 
Are there similar conformal-like transformations in 3D? Perhaps they wouldn't be called conformal maps, but maybe something exists which would work similarly for my 3D Laplace equations.

REPLY [4 votes]: (If I understand the question correctly) there is a theorem by Liouville stating that starting from dimension 3 the only conformal transformations are the (analogs of) the Möbius ones - composites of reflections in hyperspheres and hyperplanes. It is e. g. in Wikipedia. (Separate Wikipedia entry)
There are more general quasiconformal maps, maybe they are useful here, this I don't know.
PS Just notices that this has been already said in comments to the OP actually, so I probably do not deserve these upvotes...