Biharmonic maps and morphisms from conformal mappings
A biharmonic map is a smooth map between Riemannian manifolds with vanishing bitension field and in the compact case it is a critical point of the bienergy functional. Any harmonic map is biharmonic. Inspired by harmonic morphisms [\textit{P. Baird; J. C. Wood}, Harmonic morphisms between Riemannian manifolds. London Mathematical Society Monographs. New Series 29. Oxford: Oxford University Press. (2003; Zbl 1055.53049)], a continuous map between Riemannian manifolds is called a biharmonic morphism if for any biharmonic local function on the target, the pull-back function is also biharmonic. The first result here characterizes biharmonic morphisms and as a consequence, a biharmonic morphism with a 4-dimensional target always has minimal fibres, but there exist horizontally conformal submersions which are 4-harmonic, have minimal fibres but are not biharmonic morphisms. By computing the bitension field of horizontally weakly conformal maps, in 4-dimensions several examples of non-harmonic, biharmonic conformal maps are constructed, as well as a family of Riemannian submersions which are biharmonic non-harmonic maps. For certain maps between manifolds of the same dimension n, the biharmonicity condition holds if and only if \(n=4\).
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