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Tohoku Mathematical Journal
SECOND SERIES VOL. 62, NO. 1
|Tohoku Math. J.|
62 (2010), 55-73
BIHARMONIC MAPS AND MORPHISMS FROM CONFORMAL MAPPINGS
Eric Loubeau and Ye-Lin Ou
(Received June 5, 2009, revised September 29, 2009)
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.
2000 Mathematics Subject Classification.
Primary 58E20; Secondary 53C43.
Key words and phrases.
Biharmonic maps, conformal maps, biharmonic morphisms.
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