## Stable hypersurfaces with constant scalar curvature

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- by Hilário Alencar, Walcy Santos and Detang Zhou PDF
- Proc. Amer. Math. Soc.
**138**(2010), 3301-3312 Request permission

## Abstract:

We obtain some nonexistence results for complete noncompact stable hypersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb {R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int _M|A|^3<+\infty$).## References

- H. Alencar, M. do Carmo, and A. G. Colares,
*Stable hypersurfaces with constant scalar curvature*, Math. Z.**213**(1993), no. 1, 117–131. MR**1217674**, DOI 10.1007/BF03025712 - Hilário Alencar, Manfredo do Carmo, and Maria Fernanda Elbert,
*Stability of hypersurfaces with vanishing $r$-mean curvatures in Euclidean spaces*, J. Reine Angew. Math.**554**(2003), 201–216. MR**1952173**, DOI 10.1515/crll.2003.006 - Hilário Alencar, Manfredo do Carmo, and Harold Rosenberg,
*On the first eigenvalue of the linearized operator of the $r$th mean curvature of a hypersurface*, Ann. Global Anal. Geom.**11**(1993), no. 4, 387–395. MR**1246197**, DOI 10.1007/BF00773553 - Alencar, H., Santos, W., Zhou, D., Curvature integral estimates for complete hypersurfaces, arXiv:0903.2035.
- Xu Cheng,
*On constant mean curvature hypersurfaces with finite index*, Arch. Math. (Basel)**86**(2006), no. 4, 365–374. MR**2223272**, DOI 10.1007/s00013-005-1601-x - Shiu Yuen Cheng and Shing Tung Yau,
*Hypersurfaces with constant scalar curvature*, Math. Ann.**225**(1977), no. 3, 195–204. MR**431043**, DOI 10.1007/BF01425237 - Shiing-shen Chern,
*On the curvatures of a piece of hypersurface in euclidean space*, Abh. Math. Sem. Univ. Hamburg**29**(1965), 77–91. MR**188949**, DOI 10.1007/BF02996311 - Manfredo P. do Carmo and Detang Zhou,
*Eigenvalue estimate on complete noncompact Riemannian manifolds and applications*, Trans. Amer. Math. Soc.**351**(1999), no. 4, 1391–1401. MR**1451597**, DOI 10.1090/S0002-9947-99-02061-9 - Maria Fernanda Elbert,
*Constant positive 2-mean curvature hypersurfaces*, Illinois J. Math.**46**(2002), no. 1, 247–267. MR**1936088** - Maria Fernanda Elbert, Barbara Nelli, and Harold Rosenberg,
*Stable constant mean curvature hypersurfaces*, Proc. Amer. Math. Soc.**135**(2007), no. 10, 3359–3366. MR**2322768**, DOI 10.1090/S0002-9939-07-08825-9 - Haizhong Li,
*Hypersurfaces with constant scalar curvature in space forms*, Math. Ann.**305**(1996), no. 4, 665–672. MR**1399710**, DOI 10.1007/BF01444243 - R. Schoen, L. Simon, and S. T. Yau,
*Curvature estimates for minimal hypersurfaces*, Acta Math.**134**(1975), no. 3-4, 275–288. MR**423263**, DOI 10.1007/BF02392104 - Yi-Bing Shen and Xiao-Hua Zhu,
*On stable complete minimal hypersurfaces in $\textbf {R}^{n+1}$*, Amer. J. Math.**120**(1998), no. 1, 103–116. MR**1600268** - Robert C. Reilly,
*Variational properties of functions of the mean curvatures for hypersurfaces in space forms*, J. Differential Geometry**8**(1973), 465–477. MR**341351** - Harold Rosenberg,
*Hypersurfaces of constant curvature in space forms*, Bull. Sci. Math.**117**(1993), no. 2, 211–239. MR**1216008**

## Additional Information

**Hilário Alencar**- Affiliation: Instituto de Matemática, Universidade Federal de Alagoas, 57072-900 Maceió-AL, Brazil
- Email: hilario@mat.ufal.br
**Walcy Santos**- Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21941-909, Rio de Janeiro-RJ, Brazil
- Email: walcy@im.ufrj.br
**Detang Zhou**- Affiliation: Instituto de Matemática, Universidade Federal Fluminense, 24020-140, Niterói-RJ, Brazil
- Email: zhou@impa.br
- Received by editor(s): September 10, 2009
- Received by editor(s) in revised form: December 25, 2009
- Published electronically: April 22, 2010
- Additional Notes: The authors were partially supported by CNPq and FAPERJ, Brazil.
- Communicated by: Chuu-Lian Terng
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**138**(2010), 3301-3312 - MSC (2010): Primary 53C42
- DOI: https://doi.org/10.1090/S0002-9939-10-10388-8
- MathSciNet review: 2653960