Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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

  • Seo, Keom-Kyo (Department of Mathematics Sookmyung Women's University)
  • Received : 2009.07.21
  • Accepted : 2009.09.09
  • Published : 2011.03.31
Download PDF
⟨ Previous Next ⟩

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

References

  1. 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. https://doi.org/10.1007/BF02392562
  2. P. Berard and R. Sa Earp, Minimal hypersurfaces in $H^n$ ${\times}$ R, total curvature and index, arXiv: 0808.3838v1.
  3. 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. https://doi.org/10.1307/mmj/1177681991
  4. L. Hauswirth, Minimal surfaces of Riemann type in three-dimensional product manifolds, Pacific J. Math. 224 (2006), no. 1, 91-117. https://doi.org/10.2140/pjm.2006.224.91
  5. 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. https://doi.org/10.1007/BF01388843
  6. W. Jagy, Minimal hypersurfaces foliated by spheres, Michigan Math. J. 38 (1991), no. 2, 255-270. https://doi.org/10.1307/mmj/1029004332
  7. W. Jagy, Sphere-foliated constant mean curvature submanifolds, Rocky Mountain J. Math. 28 (1998), no. 3, 983-1015. https://doi.org/10.1216/rmjm/1181071750
  8. R. Lopez, Constant mean curvature hypersurfaces foliated by spheres, Differential Geom. Appl. 11 (1999), no. 3, 245-256. https://doi.org/10.1016/S0926-2245(99)00037-6
  9. B. Nelli and H. Rosenberg, Minimal surfaces in $H^2$ ${times}$ R, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263-292. https://doi.org/10.1007/s005740200013
  10. 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. https://doi.org/10.1007/s10455-007-9087-3
  11. 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. https://doi.org/10.1216/rmjm/1034968429
  12. 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.
  13. S. Stahl, The Poincare Half-Plane, Jones and Bartlett Publishers, Boston, MA, 1993.