Biharmonic maps and morphisms from conformal mappings
type: Item
instance of: scholarly article
title: Biharmonic maps and morphisms from conformal mappings
zbMATH Open document ID: 1202.53061
author: Ye-Lin Ou
author: Eric Loubeau
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2010-06-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0804.1752
review text: 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 \(n=4\).
reviewed by: Cornelia-Livia Bejan
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 5715724
zbMATH Keywords: biharmonic maps
zbMATH Keywords: conformal maps
zbMATH Keywords: biharmonic morphisms
zbMATH Keywords: bienergy functional
MaRDI profile type: Publication
OpenAlex ID: W2080871054
arXiv ID: 0804.1752
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DOI: 10.2748/TMJ/1270041027
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description: scientific article