Biharmonic maps and morphisms from conformal mappings

By Eric Loubeau, Ye-lin Ou. 2008

55 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3f1af2d10f0af4be19c67173fc46b75313f65713

Citations in the corpus

• Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces (2009, unknown): “x(2)1 + x 2 2 , x3) and a certain conformally flat metric ḡ ([27]); (iii) Biharmonic Riemannian submersions [20]: φ : (R(2) × R, dx(2) + dy(2) + β(2)(x)dz(2)) → (R(2), dx(2) + dy(2)), φ(x, y, z) = (x, y) with c1, c2 ∈ R∗, β = c2 e R f(x) , and f(x) = −c1(1+e 1) 1−ec1x , is a family of biharmonic Rie”