Biharmonic Maps from Tori into a 2-Sphere
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Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree ±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree ±1 in a large family of maps from a torus into a sphere.
KeywordsBiharmonic maps Biharmonic tori Harmonic maps Gauss maps Maps into a sphere
2000 MR Subject Classification58E20 53C12
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- Baird, P. and Wood, J. C., Harmonic Morphisms Between Riemannian Manifolds, London Math. Soc. Monogr. (N. S.), 29, Oxford Univ. Press, 2003.Google Scholar
- Eells, J. and Wood, J. C., The existence and construction of certain harmonic maps, Symposia Mathematica, Vol. XXVI (Rome, 1980), Academic Press, London, New York, 1982, 123–138.Google Scholar
- Jiang, G. Y., Some non-existence theorems of 2-harmonic isometric immersions into Euclidean spaces, Chin. Ann. Math., Ser. A, 8(3), (1987), 376–383.Google Scholar
- Nakauchi, N. and Urakawa, H., Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results. Math., 2011, DOI: 10.1007/s00025-011-0209-7.Google Scholar
- Ou, Y.-L., Some recent progress of biharmonic submanifolds, Contemporary Math. AMS, to appear, 2016.Google Scholar