# STABLE MINIMAL HYPERSURFACES IN THE HYPERBOLIC SPACE

• Seo, Keom-Kyo (DEPARTMENT OF MATHEMATICS SOOKMYUNG WOMEN'S UNIVERSITY)
• Received : 2009.09.02
• Published : 2011.03.01

#### Abstract

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.

#### Acknowledgement

Supported by : Sookmyung Women's University

#### References

1. J. Barbosa, M. Dajczer, and L. Jorge, Minimal ruled submanifolds in spaces of constant curvature, Indiana Univ. Math. J. 33 (1984), no. 4, 531-547. https://doi.org/10.1512/iumj.1984.33.33028
2. A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3567-3575. https://doi.org/10.1090/S0002-9947-07-04104-9
3. M. do Carmo and M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 (1983), no. 2, 685-709. https://doi.org/10.1090/S0002-9947-1983-0694383-X
4. M. do Carmo and C. K. Peng, Stable complete minimal hypersurfaces, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), 1349-1358, Science Press, Beijing, 1982.
5. I. Chavel, Isoperimetric Inequalities, Cambridge Tracts in Mathematics, 145. Cambridge University Press, Cambridge, 2001.
6. S. Y. Cheng, P. Li, and S.-T. Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984), no. 5, 1033-1065. https://doi.org/10.2307/2374272
7. S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, 1970 Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) pp. 59-75 Springer, New York.
8. L. F. Cheung and P. F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525-530. https://doi.org/10.1007/PL00004840
9. J. Choe, The isoperimetric inequality for minimal surfaces in a Riemannian manifold, J. Reine Angew. Math. 506 (1999), 205-214.
10. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. https://doi.org/10.1002/cpa.3160330206
11. D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727.
12. H. P. McKean, An upper bound to the spectrum of $\Delta$ on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359-366.
13. H. Mori, Minimal surfaces of revolution in $H^{3}$ and their global stability, Indiana Univ. Math. J. 30 (1981), no. 5, 787-794. https://doi.org/10.1512/iumj.1981.30.30057
14. J. Ripoll, Helicoidal minimal surfaces in hyperbolic space, Nagoya Math. J. 114 (1989), 65-75. https://doi.org/10.1017/S0027763000001409
15. L.-F. Tam and D. Zhou, Stability properties for the higher dimensional catenoid in $R^{n+1}$, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3451-3461. https://doi.org/10.1090/S0002-9939-09-09962-6
16. Y. L. Xin, Bernstein type theorems without graphic condition, Asian J. Math. 9 (2005), no. 1, 31-44. https://doi.org/10.4310/AJM.2005.v9.n1.a3

#### Cited by

1. Lpharmonic 1-forms and first eigenvalue of a stable minimal hypersurface vol.268, pp.1, 2014, https://doi.org/10.2140/pjm.2014.268.205
2. Fundamental tone of minimal hypersurfaces with finite index in hyperbolic space vol.2016, pp.1, 2016, https://doi.org/10.1186/s13660-016-1071-7
3. Vanishing theorems for $$L^{2}$$ L 2 harmonic forms on complete Riemannian manifolds vol.184, pp.1, 2016, https://doi.org/10.1007/s10711-016-0165-1
4. On the Fundamental Tone of Minimal Submanifolds with Controlled Extrinsic Curvature vol.40, pp.3, 2014, https://doi.org/10.1007/s11118-013-9349-6
5. Conformal type of ends of revolution in space forms of constant sectional curvature vol.49, pp.2, 2016, https://doi.org/10.1007/s10455-015-9484-y
6. Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature vol.41, pp.4, 2012, https://doi.org/10.1007/s10455-011-9293-x