Hi, harmonicity in 2d is preserved under mappings that satisfy the Cauchy-Riemann equations.

What about 3D? What conditions should a mapping satisfy to preserve harmonicity?

I mean, there are lots of mappings that do that; for example take any analytic function which defines a map from R2 to R2; extend it trivially to third dimension and bingo! you got yourself a harmonicity-preserving map... there are other "non-trivial" examples: for instance combining two of the above "trivial" maps can give a "non-trivial" one...

is there a general characterization a la CR for 3D?