Biharmonic maps and morphisms from conformal mappings

TL;DR

This paper explores the relationship between biharmonicity and conformality, characterizing biharmonic morphisms, especially in four dimensions, and providing new examples of proper biharmonic conformal maps and submersions.

Contribution

It provides the first example of a biharmonic morphism not of harmonic type and characterizes conditions for conformal maps to preserve biharmonicity in four dimensions.

Findings

Characterization of biharmonic morphisms.

Explicit conditions for conformal maps to preserve biharmonicity.

Examples of proper biharmonic conformal maps and submersions.

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.