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.
We prove that if an $n$-dimensional complete minimal submanifold $M$ in
hyperbolic space has sufficiently small total scalar curvature then $M$ has
only one end. We also prove that for such $M$ there exist no nontrivial $L^2$
harmonic 1-forms on $M$
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L 2 harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.Mathematics Subject Classification(2000) : 53C42, 58C40.
In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.
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