Let M be an n-dimensional complete non-compact submanifold in a hyperbolic space with the norm of its mean curvature vector bounded by a constant α < n−1. We prove in this paper that λ 1 (M ) ≥ 1 4 (n − 1 − α) 2 > 0. In particular when M is minimal we have λ 1 (M ) ≥ 1 4 (n − 1) 2 and this is sharp because equality holds when M is totally geodesic.
Mathematics Subject Classification (1991):53C20
We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal hypersurface in R n+1 , n ≥ 3, must have only one end. Any complete noncompact weakly stable CMC H-hypersurface in the hyperbolic space H n+1 , n = 3, 4, with H 2 ≥ 10 9 , 7 4 , respectively, has only one end. * Supported by CNPq of Brazil † Supported by CAPES and CNPq of Brazil.
For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.
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