STABLE MINIMAL HYPERSURFACES IN THE HYPERBOLIC SPACE
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초록
In this paper we give an upper bound of the rst eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has nite 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.
Abstract
In this paper we give an upper bound of the rst eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has nite 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.
참고문헌 (16)
- J. Barbosa, "Minimal ruled submanifolds in spaces of constant curvature", Indiana Univ. Math. J, 33(4), pp.531~547 (1984)
- A. Candel, "Eigenvalue estimates for minimal surfaces in hyperbolic space", Trans. Amer. Math. Soc, 359(8), pp.3567~3575 (2007)
- M. do Carmo, "Rotation hypersurfaces in spaces of constant curvature", Trans. Amer. Math. Soc, 277(2), pp.685~709 (1983)
- M. do Carmo, "Stable complete minimal hypersurfaces", Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, pp.1349~1358 (1982), Science Press
- I. Chavel, "Isoperimetric Inequalities, Cambridge Tracts in Mathematics" (2001), Cambridge University Press
- S. Y. Cheng, "Heat equations on minimal submanifolds and their applications", Amer. J. Math, 106(5), pp.1033~1065 (1984)
- S. S. Chern, "Minimal submanifolds of a sphere with sec- ond fundamental form of constant length, 1970 Functional Analysis and Related Fields", Proc. Conf. for M. Stone, Univ. Chicago, pp.59~75 (1968), Springer
- L. F. Cheung, "Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space", Math. Z, 236(3), pp.525~530 (2001)
- J. Choe, "The isoperimetric inequality for minimal surfaces in a Riemannian manifold", J. Reine Angew. Math, 506, pp.205~214 (1999)
- D. Fischer-Colbrie, "The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature", Comm. Pure Appl. Math, 33(2), pp.199~211 (1980)
- D. Hoffman, "Sobolev and isoperimetric inequalities for Riemannian sub- manifolds", Comm. Pure Appl. Math, 27, pp.715~727 (1974)
- H. P. McKean, "An upper bound to the spectrum of Δ on a manifold of negative curvature", J. Differential Geometry, 4, pp.359~366 (1970)
- H. Mori, "Minimal surfaces of revolution in H3 and their global stability", Indiana Univ. Math. J, 30(5), pp.787~794 (1981)
- J. Ripoll, "Helicoidal minimal surfaces in hyperbolic space", Nagoya Math. J, 114, pp.65~75 (1989)
- L.-F. Tam, "Stability properties for the higher dimensional catenoid in Rn+1", Proc. Amer. Math. Soc, 137(10), pp.3451~3461 (2009)
- Y. L. Xin, "Bernstein type theorems without graphic condition", Asian J. Math, 9(1), pp.31~44 (2005)