Biharmonic Maps from Tori into a 2-Sphere

Citation:

Zeping WANG,Ye-Lin OU,Hanchun YANG.Biharmonic Maps from Tori into a 2-Sphere[J].Chinese Annals of Mathematics B,2018,39(5):861~878
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Authors:

Zeping WANG; Ye-Lin OU;Hanchun YANG

Foundation:

The third author was supported by the Natural Science Foundation of China (No.11361073) and the second author was supported by the Natural Science Foundation of Guangxi Province of China (No.2011GXNSFA018127).
Abstract: 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 $\pm 1$. 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 $\pm 1$ in a large family of maps from a torus into a sphere.

Keywords:

Biharmonic maps, Biharmonic tori, Harmonic maps, Gauss maps, Maps into a sphere

Classification:

58E20, 53C12
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