Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The $ r$-stability of hypersurfaces with zero Gauss-Kronecker curvature

Author: Marcos P. A. Cavalcante
Journal: Proc. Amer. Math. Soc. 136 (2008), 287-294
MSC (2000): Primary 53C42, 53A07; Secondary 35P15
Published electronically: September 24, 2007
MathSciNet review: 2350415
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give sufficient conditions for a bounded domain in an $ r$-minimal hypersurface of the Euclidean space to be $ r$-stable. The Gauss-Kronecker curvature of this hypersurface may be zero on a set of capacity zero.

References [Enhancements On Off] (What's this?)

  • 1. H. Alencar, M. do Carmo and M. F. Elbert,
    Stability of hypersurfaces with vanishing $ r$-mean curvatures in Euclidean spaces.
    J. reine angew. Math. 554 (2003), 201-216. MR 1952173 (2003k:53061)
  • 2. J. L. M. Barbosa and A. G. Colares,
    Stability of hypersurfaces with constant $ r$-mean curvature.
    Ann. Global Anal. Geom. 15 (1997), no. 3, 277-297. MR 1456513 (98h:53091)
  • 3. J. L. M. Barbosa and M. P. do Carmo,
    On the size of a stable minimal surface in $ R^3$.
    Amer. J. Math. 98(1976), 515-528. MR 0413172 (54:1292)
  • 4. G. Courtois,
    Spectrum of manifolds with holes.
    J. Funct. Anal. 134 (1995), no. 1, 194-221. MR 1359926 (97b:58142)
  • 5. L. C. Evans and R. F. Gariepy,
    Measure theory and fine properties of functions.
    Studies in Advanced Mathematics (1992), 268 p. MR 1158660 (93f:28001)
  • 6. D. Fischer-Colbrie and R. Schoen,
    The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvatures.
    Comm. Pure Appl. Math. 33 (1980), 199-211. MR 562550 (81i:53044)
  • 7. J. Hounie and M. L. Leite,
    The maximum principle for hypersurfaces with vanishing curvature functions.
    J. Diff. Geom. 41 (1995), 247-258. MR 1331967 (96b:53080)
  • 8. J. Rauch and M. Taylor,
    Potential and scattering theory on wildly perturbed domains,
    J. Funct. Anal. 18 (1975), 27-59. MR 0377303 (51:13476)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 53A07, 35P15

Retrieve articles in all journals with MSC (2000): 53C42, 53A07, 35P15

Additional Information

Marcos P. A. Cavalcante
Affiliation: Instituto de Matemática, Universidade Federal de Alagoas, Campus A. C. Simões, BR 104, Norte, Km 97, 57072-970, Maceió, AL, Brazil

Keywords: $r$-minimal immersions, $r$-stability, capacity.
Received by editor(s): April 20, 2006
Received by editor(s) in revised form: September 22, 2006
Published electronically: September 24, 2007
Additional Notes: The author was fully supported by CNPq-Brazil.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society