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. nonharmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps.

Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

“…We refer the readers to the book [2] for a comprehensive account of the theory, applications and interesting links of harmonic morphisms. Biharmonic morphisms, as a generalization of the notion of harmonic morphisms, were introduced and studied in [9], [5] and [6]. These are maps between Riemannian manifolds which preserve the solutions of bi-Laplace equations in the sense that they pull back germs of biharmonic functions to germs of biharmonic functions.…”

“…These are maps between Riemannian manifolds which preserve the solutions of bi-Laplace equations in the sense that they pull back germs of biharmonic functions to germs of biharmonic functions. According to a characterization obtained in [6], a map between Riemannian manifolds is a biharmonic morphism if and only if it is a horizontally weakly conformal map which is also a biharmonic map, a 4-harmonic map, and satisfies an additional equation. So biharmonic morphisms are a very restricted class of horizontally weakly conformal biharmonic maps.…”

On conformal biharmonic immersions

“…However, apart from the maps between Euclidean spaces defined by polynomials of degree less than four (a class of maps that seems so wild to exhibit any characteristic property) not many examples of proper biharmonic maps between Riemannain manifolds have been found (see, e.g., [15], [14], [17], and the bibliography of biharmonic maps [12]). So, currently, one priority and a practical thing to do seems to be finding more examples of proper biharmonic maps between certain model spaces or studying biharmonic maps under some geometric constraints.…”

“…So, currently, one priority and a practical thing to do seems to be finding more examples of proper biharmonic maps between certain model spaces or studying biharmonic maps under some geometric constraints. For example, one can study biharmonic isometric immersions which lead to the concept of biharmonic submanifolds (see e.g., [11], [6], [7], [4], [5], [15] and [3]); one can also study, as in [1], [2], [14], horizontally weakly conformal biharmonic maps which generalize both the notion of harmonic morphisms (maps that are both horizontally weakly conformal and harmonic) and that of biharmonic morphisms (maps that are horizontally weakly conformal biharmonic with other constraints, see [16], [13], [17], and [14] for details).…”

Harmonic Maps and Biharmonic Maps

“…In the case that a Riemannian submersion from a space form of constant sectional curvature into a Riemann surface (N 2 , h), Wang and Ou (cf., [19,28]) showed that it is biharmonic if and only if it is harmonic. We treat with a submersion from a higher dimensional Riemannian manifold (M, g) (cf., [51]).…”