Some classifications of conformal biharmonic and k-polyharmonic maps

We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a M\"obius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of M\"obius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.