Arkiv för Matematik

  • Ark. Mat.
  • Volume 54, Number 2 (2016), 233-241.

Stable hypersurfaces with zero scalar curvature in Euclidean space

Hilário Alencar, Manfredo Carmo, and Gregório Silva Neto

Full-text: Open access


In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.


Hilário Alencar and Manfredo do Carmo were partially supported by CNPq of Brazil

Article information

Ark. Mat., Volume 54, Number 2 (2016), 233-241.

Received: 25 September 2015
Revised: 23 January 2016
First available in Project Euclid: 30 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2016 © Institut Mittag-Leffler


Alencar, Hilário; Carmo, Manfredo; Neto, Gregório Silva. Stable hypersurfaces with zero scalar curvature in Euclidean space. Ark. Mat. 54 (2016), no. 2, 233--241. doi:10.1007/s11512-016-0232-8.

Export citation


  • Alencar, H., do Carmo, M. and Elbert, M. F., Stability of hypersurfaces with vanishing r-mean curvatures in Euclidean spaces, J. Reine Angew. Math. 554 (2003), 201–216.
  • Alencar, H., Santos, W. and Zhou, D., Stable hypersurfaces with constant scalar curvature, Proc. Amer. Math. Soc. 138 (2010), 3301–3312.
  • Elbert, M. F., Constant positive 2-mean curvature hypersurfaces, Illinois J. Math. 46 (2002), 247–267.
  • Hounie, J. and Luiza Leite, M., Two-ended hypersurfaces with zero scalar curvature, Indiana Univ. Math. J. 48 (1999), 867–882.
  • Hounie, J. and Luiza Leite, M., The maximum principle for hypersurfaces with vanishing curvature functions, J. Differential Geom. 41 (1995), 247–258.
  • Reilly, R. C., Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom. 8 (1973), 465–477.
  • Neto, G. S., On stable hypersurfaces with vanishing scalar curvature, Math. Z. 277 (2014), 481–497.
  • Smale, S. J., On Morse index theorem, J. Math. Mech. 14 (1965), 1049–1055.