ABSTRACT

The periodic *s*^{1}-equivariant hypersurfaces of constant mean curvature can be obtained by using the Lagrangians with suitable potential functions in the Berger spheres. In the corresponding Hamiltonian system, the conservation law is effectively applied to the construction of periodic *s*^{1}-equivariant surfaces of arbitrary positive constant mean curvature.

Cite this paper

K. Kikuchi, "*S*^{1}-Equivariant CMC Surfaces in the Berger Sphere and the Corresponding Lagrangians," *Advances in Pure Mathematics*, Vol. 3 No. 2, 2013, pp. 259-263. doi: 10.4236/apm.2013.32037.

K. Kikuchi, "

References

[1] W-Y. Hsiang, “On Generalization of Theorems of A. D. Alexandrov and C. Delaunay on Hypersurfaces of Constant Mean Curvature,” Duke Mathematical Journal, Vol. 49, No. 3, 1982, pp. 485-496. doi:10.1215/S0012-7094-82-04927-4

[2] J. Eells and A. Ratto, “Harmonic Maps and Minimal Immersions with Symmetries,” Annals of Mathematics Stu- dies, No. 130, 1993.

[3] K. Kikuchi, “The Construction of Rotation Surfaces of Constant Mean Curvature and the Corresponding Lagrangians,” Tsukuba Journal of Mathematics, Vol. 36, No. 1, 2012, pp. 43-52.

[4] W-Y. Hsiang and H. B. Lawson, “Minimal Submanifolds of Low Cohomogeneity,” Journal of Differential Geometry, Vol. 5, 1971, pp. 1-38.

[5] D. Ferus and U. Pinkall, “Constant Curvature 2-Spheres in the 4-Sphere,” Mathematische Zeitschrift, Vol. 200, No. 2, 1989, pp. 265-271. doi:10.1007/BF01230286

[6] H. Muto, Y. Ohnita and H. Urakawa, “Homogeneous Minimal Hypersurfaces in the Unit Spheres and the First Eigenvalues of Their Laplacian,” Tohoku Mathematical Journal, Vol. 36, No. 2, 1984, pp. 253-267. doi:10.2748/tmj/1178228851

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[1] W-Y. Hsiang, “On Generalization of Theorems of A. D. Alexandrov and C. Delaunay on Hypersurfaces of Constant Mean Curvature,” Duke Mathematical Journal, Vol. 49, No. 3, 1982, pp. 485-496. doi:10.1215/S0012-7094-82-04927-4

[2] J. Eells and A. Ratto, “Harmonic Maps and Minimal Immersions with Symmetries,” Annals of Mathematics Stu- dies, No. 130, 1993.

[3] K. Kikuchi, “The Construction of Rotation Surfaces of Constant Mean Curvature and the Corresponding Lagrangians,” Tsukuba Journal of Mathematics, Vol. 36, No. 1, 2012, pp. 43-52.

[4] W-Y. Hsiang and H. B. Lawson, “Minimal Submanifolds of Low Cohomogeneity,” Journal of Differential Geometry, Vol. 5, 1971, pp. 1-38.

[5] D. Ferus and U. Pinkall, “Constant Curvature 2-Spheres in the 4-Sphere,” Mathematische Zeitschrift, Vol. 200, No. 2, 1989, pp. 265-271. doi:10.1007/BF01230286

[6] H. Muto, Y. Ohnita and H. Urakawa, “Homogeneous Minimal Hypersurfaces in the Unit Spheres and the First Eigenvalues of Their Laplacian,” Tohoku Mathematical Journal, Vol. 36, No. 2, 1984, pp. 253-267. doi:10.2748/tmj/1178228851

[7] P. Petersen, “Riemannian Geometry,” Graduate Texts in Mathematics, 2nd Edition, Vol. 171, Springer-Verlag, New York, 2006.