Tohoku Mathematical Journal

On stable complete hypersurfaces with vanishing {$r$}-mean curvature

Maria F. Elbert and Manfredo do Carmo

Full-text: Open access

Abstract

A form of Bernstein theorem states that a complete stable minimal surface in euclidean space is a plane. A generalization of this statement is that there exists no complete stable hypersurface of an $n$-euclidean space with vanishing $(n-1)$-mean curvature and nowhere zero Gauss-Kronecker curvature. We show that this is the case, provided the immersion is proper and the total curvature is finite.

Article information

Source
Tohoku Math. J. (2), Volume 56, Number 2 (2004), 155-162.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246548

Digital Object Identifier
doi:10.2748/tmj/1113246548

Mathematical Reviews number (MathSciNet)
MR2053316

Zentralblatt MATH identifier
1062.53052

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
Stability r-mean curvature complete finite total curvature

Citation

do Carmo, Manfredo; Elbert, Maria F. On stable complete hypersurfaces with vanishing {$r$}-mean curvature. Tohoku Math. J. (2) 56 (2004), no. 2, 155--162. doi:10.2748/tmj/1113246548. https://projecteuclid.org/euclid.tmj/1113246548


Export citation

References

  • 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.
  • J. L. Barbosa and M. P. do Carmo, On the size of a stable minimal surface in $\re^3$, Amer. J. Math. 98 (1976), 515--528.
  • J. L. Barbosa and A. G. Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), 277--297.
  • M. do Carmo and A. da Silveira, Globally stable complete minimal surfaces in $\re^3$, Proc. Amer. Math. Soc. 79 (1980), 345--346.
  • M. do Carmo and C. K. Peng, Stable complete minimal surfaces in $\re^3$ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 903--906.
  • M. do Carmo and M. F. Elbert, Complete hypersurfaces in Euclidean spaces with finite total curvature, Preprint, (2003).
  • D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces on 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199--211.
  • J. Hounie and M. L. Leite, The maximum principle for hypersurfaces with vanishing curvature functions, J. Differential Geom. 41 (1995), 247--258.
  • J. Hounie and M. L. Leite, Two-ended hypersurfaces with zero scalar curvature, Indiana Univ. Math. J. 48 (1999), 867--882.
  • J. Hounie and M. L. Leite, Uniqueness and non-existence theorems for hypersurfaces with $H_r=0$, Ann. Global Anal. Geom. 17 (1999), 397--407.
  • R. Reilly, Variational properties of functions of the mean curvature in space forms, J. Differential Geom. 8 (1973), 465--477.
  • H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), 211--239.
  • T. Sakai, Riemannian Geometry, Transl. Math. Monogr. 149, American Mathematical Society, Providence, RI, 1996.
  • M. Traizet, On the stable surface of constant Gauss curvature in space forms, Ann. Global Anal. Geom. 13 (1995), 141--148.