Some classifications of conformal biharmonic and k-polyharmonic maps

TL;DR

This paper classifies conformal biharmonic and k-polyharmonic maps between space forms, revealing that proper biharmonic maps occur only in specific dimensions and are related to M"obius transformations, providing new explicit examples.

Contribution

It provides a complete classification of conformal biharmonic maps between space forms and characterizes proper k-polyharmonic conformal maps, linking them to M"obius transformations.

Findings

Proper biharmonic conformal maps occur only in 4-dimensional flat space forms.

Proper k-polyharmonic conformal maps exist only in dimension 2k and are restrictions of M"obius transformations.

The paper offers explicit examples of proper k-polyharmonic maps with geometric significance.

Abstract

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.