Tohoku Mathematical Journal

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., 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.

Article information

Source
Tohoku Math. J. (2), Volume 62, Number 1 (2010), 55-73.

Dates
First available in Project Euclid: 31 March 2010

https://projecteuclid.org/euclid.tmj/1270041027

Digital Object Identifier
doi:10.2748/tmj/1270041027

Mathematical Reviews number (MathSciNet)
MR2654303

Zentralblatt MATH identifier
1202.53061

Citation

Loubeau, Eric; Ou, Ye-Lin. Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. (2) 62 (2010), no. 1, 55--73. doi:10.2748/tmj/1270041027. https://projecteuclid.org/euclid.tmj/1270041027

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