[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2011.48.2.335

SPHERE-FOLIATED MINIMAL AND CONSTANT MEAN CURVATURE HYPERSURFACES IN PRODUCT SPACES

Seo, Keom-Kyo (Department of Mathematics Sookmyung Women's University)

Publication Information

Bulletin of the Korean Mathematical Society / v.48, no.2, 2011 , pp. 335-342 More about this Journal

Abstract

In this paper, we prove that minimal hypersurfaces when n $\geq$ 3 and nonzero constant mean curvature hypersurfaces when n $\geq$ 2 foliated by spheres in parallel horizontal hyperplanes in $\mathbb{H}^n{\times}\mathbb{R}$ must be rotationally symmetric.

Keywords

foliation;constant mean curvature;rotationally symmetric hypersurface;product space;

Citations & Related Records

Times Cited By Web Of Science : 0 (Related Records In Web of Science)
Times Cited By SCOPUS : 0

Reference

1	W.-T. Hsiang and W.-Y. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math. 98 (1989), no. 1, 39-58. DOI
2	W. Jagy, Minimal hypersurfaces foliated by spheres, Michigan Math. J. 38 (1991), no. 2, 255-270. DOI
3	W. Jagy, Sphere-foliated constant mean curvature submanifolds, Rocky Mountain J. Math. 28 (1998), no. 3, 983-1015. DOI
4	R. Lopez, Constant mean curvature hypersurfaces foliated by spheres, Differential Geom. Appl. 11 (1999), no. 3, 245-256. DOI ScienceOn
5	B. Nelli and H. Rosenberg, Minimal surfaces in $H^2$ ${times}$ R, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263-292. DOI ScienceOn
6	B. Nelli, R. Sa Earp, W. Santos, and E. Toubiana, Uniqueness of H-surfaces in $H^2$ ${\times}$ R, ${\left\|H\right\|}$ ${\leq}$ 1/2, with boundary one or two parallel horizontal circles, Ann. Global Anal. Geom. 33 (2008), no. 4, 307-321. DOI ScienceOn
7	S.-H. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski space, Rocky Mountain J. Math. 32 (2002), no. 3, 1019-1044. DOI ScienceOn
8	R. Sa Earp and E. Toubiana, Screw motion surfaces in $H^2$ ${\times}$ R and $S^2$ ${\times}$ R, Illinois J. Math. 49 (2005), no. 4, 1323-1362.
9	S. Stahl, The Poincare Half-Plane, Jones and Bartlett Publishers, Boston, MA, 1993.
10	U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in $S^2$ ${\times}$ R and $H^2$ ${\times}$ R, Acta Math. 193 (2004), no. 2, 141-174. DOI
11	P. Berard and R. Sa Earp, Minimal hypersurfaces in $H^n$ ${\times}$ R, total curvature and index, arXiv: 0808.3838v1.
12	M. Cavalcante and J. de Lira, Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces, Michigan Math. J. 55 (2007), no. 1, 163-181. DOI
13	L. Hauswirth, Minimal surfaces of Riemann type in three-dimensional product manifolds, Pacific J. Math. 224 (2006), no. 1, 91-117. DOI