Question

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.

Answer 1

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 $T_xf:T_x \mathbb R^3 = \mathbb R^3\to T_{f(x)} \mathbb R^3 = \mathbb R^3$.