## Tohoku Mathematical Journal

### On stable constant mean curvature hypersurfaces

#### Abstract

We study complete non-compact stable constant mean curvature hypersurfaces in a Riemannian manifold of bounded geometry, and prove that there are no nontrivial $L^2$ harmonic 1-forms on such hypersurfaces. We also show that any smooth map with finite energy from such a hypersurface to a compact manifold with non-positive sectional curvature is homotopic to constant on each compact set. In particular, we obtain some one-end theorems of complete non-compact weakly stable constant mean curvature hypersurfaces in the space forms.

#### Article information

Source
Tohoku Math. J. (2), Volume 62, Number 3 (2010), 383-392.

Dates
First available in Project Euclid: 15 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1287148618

Digital Object Identifier
doi:10.2748/tmj/1287148618

Mathematical Reviews number (MathSciNet)
MR2742015

Zentralblatt MATH identifier
1206.53062

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 58E20: Harmonic maps [See also 53C43], etc.

#### Citation

Fu, Hai-Ping; Li, Zhen-Qi. On stable constant mean curvature hypersurfaces. Tohoku Math. J. (2) 62 (2010), no. 3, 383--392. doi:10.2748/tmj/1287148618. https://projecteuclid.org/euclid.tmj/1287148618

#### References

• M. T. Anderson, The compactification of a minimal submanifold in Euclidean space by the Gauss map, preprint (final version in Dept. of Math., California Institute of Technology, Pasadena, CA 91125), 1986.
• J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), 339--353.
• J. L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), 123--138.
• H. D. Cao, Y. Shen and S. Zhu, The structure of stable minimal hypersurfaces in $\mathbbR^n+1$, Math. Res. Lett. 4 (1997), 637--644.
• G. Carron, $L^2$ harmonic forms on non compact manifolds, arXiv:0704.3194vl.
• X. Cheng, $L^2$ harmonic forms and stability of hypersurfaces with constant mean curvature, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), 225--239.
• X. Cheng, On constant mean curvature hypersurfaces with finite index, Arch. Math. (Basel) 86 (2006), 365--374.
• X. Cheng, L. F. Cheung and D. T. Zhou, The structure of weakly stable constant mean curvature hypersufaces, Tohoku Math. J. 60 (2008), 101--121.
• A. M. Da Silveira, Stability of complete noncompact surfaces with constant mean curvature, Math. Ann. 277 (1987), 629--638.
• M. do Carmo and C. K. Peng, Stable complete minimal surfaces in $R^3$ are planes, Bull. Amer. Math. Soc. 1 (1979), 903--906.
• J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109--160.
• H. P. Fu and Z. Q. Li, $L^2$ harmonic 1-forms on complete submanifolds in Euclidean space, Kodai Math. J. 32 (2009), 432--441.
• D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure. Appl. Math. 27 (1974), 715--727.
• P. Li, Curvature and function theory on Riemannian manifolds, In Honor of Atiyah, Bott, Hirzebruch, and Singer, 375--432, Survey in Differential Geometry vol. VII, International Press, Cambridge, 2000.
• P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), 359--383.
• P. Li and J. P. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett. 9 (2002), 95--103.
• R. Miyaoka, $L^2$ harmonic $1$-forms on a complete stable minimal hypersurfaces, Geometry and Global Analysis (Sendai 1993),Tohoku Univ., 289--293.
• B. Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991), 185--188.
• R. Schoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), 333--341.
• Y. B. Shen and X. H. Zhu, On complete hypersurfaces with constant mean curvature and finite $L^p$-norm curvature in $\mathbbR^n+1$, Acta Math. Sin. (Engl. Ser.) 21 (2005), 631--642.
• K. Shiohama and H. W. Xu, The topological sphere theorem for complete submanifolds, Compositio Math. 107 (1997), 221--232.
• S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659--670.
• G. Yun, Total scalar curvature and $L^2$ harmonic 1-forms on a minimal hypersurface in Euclidean space, Geom. Dedicata 89 (2002), 135--141.