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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.

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Well, there are conformal map in higher dimensions (defined by saying that they preserve angles). However, they are quite a bit less flexible than in 2d; see my answer to the following question : mathoverflow.net/questions/10066/… –  Andy Putman Jan 16 '11 at 4:54
Can I requestion "complicated" rather than "complex" in the second paragraph? The word "complex" has all sorts of technical meaning. –  Theo Johnson-Freyd Jan 16 '11 at 8:12
Theo, you're right.. done! –  MathGeek Jan 16 '11 at 8:46
Andy, yes, those affine transforms aren't useful for this. So the question stands.. are there other transforms that DO work in 3D for the Laplace equation? Or is that really EQUIVALENT to a conformal map, meaning that only affine transforms have that characteristic in 3D? –  MathGeek Jan 16 '11 at 8:48
@MathGeek: note that higher-dimensional conformal maps are not all affine, they are slightly more general than that (restriction of Möbius transform on the sphere), but this does not help for your original question. –  Benoît Kloeckner Sep 3 '11 at 6:53

3 Answers 3

I don't know if this is relevant for your question...

In http://arxiv.org/pdf/1005.5464v2, the author introduces a notion of "weak conformal map" for 3-dimensional domains, and proves a Riemann mapping theorem for those kinds of maps.

Definition: given two open subsets $U,V\subset \mathbb R^3$, a smooth map $f:U\to V$ is called weak conformal if, at every $x\in U$, the three eigenvalues of $P_x:\mathbb R^3\to \mathbb R^3$ are in geometric progression. Here, the positive operator $P_x$ is the one coming from the polar decomposition of the tangent map $T_xf:T_x \mathbb R^3 = \mathbb R^3\to T_{f(x)} \mathbb R^3 = \mathbb R^3$.

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(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...

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Your problem is a 'classic' of evaluation of numerical methods. I presume the analytic answer is know. Of course the 3D conformal mapping of your problem exist. Notably, solving the Laplacian problem of potential! :\ I used the concept to map very complex bodies in an sphere. (For fun). Of course of the same topology. I recommend to you "Calculation of potential flow about arbitrary bodies" Hess & Smith (1969?) and Kellogg "foundation of potential theory" (1929) (if you have a lot of time!). Remember. If you follow the conditions for a Laplacian fluid you can find (I do not how analytically) a mapping. it's a very beautiful problem. The natural answer exist and it mock us. A classic mathematical problem like "It shows that if you can transform a duck and an orange the problem has a solution and you have found it". Now. Again a SQUARE domain? What is the joke?

Maybe you could try a somewhat different problem. To transform a cylinder inside other in a sphere inside other sphere. Now, you have a 2D problem. Singularity even exist (I assume you want to try some numerical procedure) and the problem can be mapped. If your problem is to evaluate a gridded domain method (like FD, FV or FE). maybe you can try an integral method. Of course. There is a problem in the singular point but if aren't nonlinearities the method is powerful.

Sorry for my very bad English. Here is 3.30am and I just have finish my drink. And. I'm not a mathematician. Only an unemployed aircraft mechanic.

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