Géométrie différentielle
Un analogue conforme des applications harmoniques
[A conformal analogue of harmonic maps]
Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 985-988.

On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is even dimensional. We then build a conformal invariant functional for the maps between two Riemannian manifolds. Its critical points then called C-harmonic are the solutions of a nonlinear elliptic PDE of order n, which is conformal invariant with respect to the start manifold. For the trivial case of real or complex functions of M, we find again the GJMS operator, with a leading part power to the n/2 of the Laplacian.

Sur une surface de Riemann, l'énergie d'une application à valeurs dans une variété riemannienne est une fonctionnelle invariante conforme, ses points critiques sont les applications harmoniques. Nous proposons ici un analogue en dimension supérieure, en construisant une fonctionnelle invariante conforme pour les applications entre deux variétés riemanniennes, dont la source est de dimension n paire. Ses points critiques satisfont une EDP elliptique d'ordre n non linéaire qui est invariante conforme sur la source, on les appelle les applications C-harmoniques. Dans le cas des fonctions, on retrouve l'opérateur GJMS, dont le terme principal est une puissance n/2 du laplacien.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.06.008
Bérard, Vincent 1

1 IRMA, CNRS et Université Louis-Pasteur, 7, rue René-Descartes, 67084 Strasbourg cedex, France
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Bérard, Vincent. Un analogue conforme des applications harmoniques. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 985-988. doi : 10.1016/j.crma.2008.06.008. http://www.numdam.org/articles/10.1016/j.crma.2008.06.008/

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