Differential Geometry
A construction of conformal-harmonic maps
Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 967-970.

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells–Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky–Viaclovsky and Lamm.

Les applications conformes-harmoniques dʼune variété conforme de dimension 4 vers une variété riemannienne sont les solutions dʼune équation non linéaire, conformément invariante, dʼordre 4. Nous démontrons un résultat général dʼexistence pour ces applications conformes-harmoniques, analogue au théorème dʼEells–Sampson pour les applications harmoniques. La démonstration utilise un flot géométrique et sʼappuie sur des résultats de Gursky–Viaclovsky et Lamm.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2012.10.021
Biquard, Olivier 1, 2; Madani, Farid 3

1 UPMC, Université Paris 6, France
2 École normale supérieure, 45 rue dʼUlm, 75005 Paris, France
3 Fakultät für Mathematik, Universität Regensburg, Germany
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Biquard, Olivier; Madani, Farid. A construction of conformal-harmonic maps. Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 967-970. doi : 10.1016/j.crma.2012.10.021. http://www.numdam.org/articles/10.1016/j.crma.2012.10.021/

[1] Bérard, V. Un analogue conforme des applications harmoniques, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 985-988

[2] Eells, J.; Sampson, J. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160

[3] Gursky, M.J.; Viaclovsky, J.A. A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom., Volume 63 (2003), pp. 131-154

[4] Lamm, T. Biharmonic map heat flow into manifolds of nonpositive curvature, Calc. Var. Partial Differential Equations, Volume 22 (2005), pp. 421-445

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