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Stable constant mean curvature hypersurfaces

Authors: Maria Fernanda Elbert, Barbara Nelli and Harold Rosenberg
Journal: Proc. Amer. Math. Soc. 135 (2007), 3359-3366
MSC (2000): Primary 53C42
Published electronically: June 19, 2007
MathSciNet review: 2322768
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Abstract: Let $ {\mathcal N}^{n+1}$ be a Riemannian manifold with sectional curvatures uniformly bounded from below. When $ n=3,4,$ we prove that there are no complete (strongly) stable $ H$-hypersurfaces, without boundary, provided $ \vert H\vert$ is large enough. In particular, we prove that there are no complete strongly stable $ H$-hypersurfaces in $ \mathbb{R}^{n+1}$ without boundary, $ H\not=0.$

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  • 1. J.L. BARBOSA, P. BÉRARD: Eigenvalue and ``Twisted'' Eigenvalue Problems, Applications to CMC Surfaces, J. Math. Pures Appl. 79, 5 (2000) 427-450. MR 1759435 (2001f:58064)
  • 2. J.L. BARBOSA, M. DO CARMO: Stability of Hypersurfaces with Constant Mean Curvature, Math. Zeit. 185, 3 (1984) 339-353. MR 0731682 (85k:58021c)
  • 3. X. CHENG: On Constant Mean Curvature Hypersurfaces with Finite Index, Preprint, to be published in Archiv der Mathematik.
  • 4. M. DO CARMO, C.K. PENG: Stable Complete Minimal Hypersurfaces, Proceedings of the 1980 Beijing Symposium on Diff. Geom. and Diff. Eq., Science Press, Beijing, (1982) 1349-1358. MR 0714373 (85e:53007)
  • 5. R. F. DELIMA, W. MEEKS, III: Maximum Principles at Infinity for Surfaces of Bounded Mean Curvature in $ \mathbb{R}^3$ and $ \mathbb{H}^3,$ Indiana Univ. Math. Journal 53, 5 (2004) 1211-1223. MR 2104275 (2005h:53011)
  • 6. D. FISCHER-COLBRIE: On Complete Minimal Surfaces with finite Morse Index in three Manifolds, Inv. Math. 82 (1985) 121-132. MR 0808112 (87b:53090)
  • 7. B. NELLI, H. ROSENBERG: Global Properties of Constant Mean Curvature Surfaces in $ \mathbb{H}^2\times\mathbb{R}$, to appear in Pacific Jour. of Math. (2004),$ \thicksim$rosen.
  • 8. H. ROSENBERG: Constant Mean Curvature Surfaces in Homogeneously Regular 3-Manifold, Preprint (2005)$ \thicksim$ rosen/. MR 1483380
  • 9. A. ROS, H. ROSENBERG: Properly Embedded Surfaces with Constant Mean Curvature, Preprint (2001),$ \thicksim$rosen.
  • 10. R. SCHOEN, S.T. YAU: Lectures on Differential Geometry, International Press, (1994). MR 1333601 (97d:53001)
  • 11. Y. SHEN, S. ZHU: Rigidity of Stable Minimal Hypersurfaces, Math. Annalen 399 (1997) 107-116. MR 1467649 (98g:53113)
  • 12. Y.B. SHEN, X.H. ZHU: On Complete Hypersurfaces with Constant Mean Curvature and Finite $ L^p$-norm Curvature in $ \mathbb{R}^{n+1},$ Acta Math. Sinica, English series 21, 3 (2005) 631-642. MR 2145917 (2006f:53088)

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Additional Information

Maria Fernanda Elbert
Affiliation: Instituto de Matematica, Universidade Federal do Rio de Janeiro, Rio de Janiero, Brazil

Barbara Nelli
Affiliation: Dipartimento di Matematica Pura e Applicata, Universitá di L’Aquila, Via Vetoio, 67010 Coppito L’Aquila, Italy

Harold Rosenberg
Affiliation: Institut de Mathématiques, Université Paris VII, 2 place Jussieu, 75251 Paris, France

Received by editor(s): January 24, 2006
Received by editor(s) in revised form: May 17, 2006
Published electronically: June 19, 2007
Additional Notes: The first author was partially supported by CNPq and Faperj.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society

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