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.
Accepted:
Published online:
Olivier Biquard 1, 2; Farid Madani 3
@article{CRMATH_2012__350_21-22_967_0, author = {Olivier Biquard and Farid Madani}, title = {A construction of conformal-harmonic maps}, journal = {Comptes Rendus. Math\'ematique}, pages = {967--970}, publisher = {Elsevier}, volume = {350}, number = {21-22}, year = {2012}, doi = {10.1016/j.crma.2012.10.021}, language = {en}, }
Olivier Biquard; Farid Madani. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.021/
[1] Un analogue conforme des applications harmoniques, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 985-988
[2] Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160
[3] A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom., Volume 63 (2003), pp. 131-154
[4] Biharmonic map heat flow into manifolds of nonpositive curvature, Calc. Var. Partial Differential Equations, Volume 22 (2005), pp. 421-445
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