Biharmonic maps and morphisms from conformal mappings
Abstract
Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly conformal maps, which include conformal mappings. This leads to several examples of proper (i.e. non-harmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps.
Cite
@article{arxiv.0804.1752,
title = {Biharmonic maps and morphisms from conformal mappings},
author = {E. Loubeau and Y. -L. Ou},
journal= {arXiv preprint arXiv:0804.1752},
year = {2008}
}