Abstract
We study a version of the Gauss map \(g:M^2 \to S^2\) for a surface \(M^2\) immersed in \(S^3\) and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff \(M^2\) has a constant mean curvature. As a corollary, we conclude that an embedded flat torus \(T^2\) with constant mean curvature is a spherical Delonay surface.
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REFERENCES
E. A. Ruh and J. Vilms, “The tension field of Gauss map,” Trans. Amer. Math. Soc., 199 (1970), 569–573.
A. A. Borisenko and Yu. A. Nikolaevskii, “Grassmannian manifolds and the Grassmannian image of submanifolds,” Uspekhi Mat. Nauk [Russian Math. Surveys], 46 (1991), 41–83.
Xiabo Liu, “Rigidity of the Gauss map in compact Lie group,” Duke Math. J., 77 (1995), no. 2, 447–480.
R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York–London, 1964.
H. Urakawa, Calculus of Variations and Harmonic Maps, Translations Math. Monographs, vol. 132, Amer. Math. Soc., Providence, R.I., 1993.
H. B. Lawson, “Complete minimal surfaces in S 3,” Ann. Math., 92 (1970), 335–374.
P. Scott, “The geometries of 3-manifolds,” The Bulletin of the London Math. Soc., 15 (1983), no. 56, 401–487.
H. Fujimoto, “Modified defect relations for the Gauss map of minimal surfaces,” J. Diff. Geom., 29 (1989), 245–262.
H. C. Wente, Constant Mean Curvature Immersions of Enneper Type, Memoirs of Amer. Math. Soc., vol. 478, Providence, R.I., 1992.
B. Solomon, “Harmonic maps to spheres,” J. Diff. Geom., 21 (1985), no. 2, 151–162.
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Masal'tsev, L.A. A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3 . Mathematical Notes 73, 85–96 (2003). https://doi.org/10.1023/A:1022126101717
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DOI: https://doi.org/10.1023/A:1022126101717