Rigidity of Immersed Submanifolds in a Hyperbolic Space

Abstract

Abstract. Let M n , 2 ≤ n ≤ 6 be a complete noncompact hypersurface immersed in H n+1 . We show that there exist two certain positive constants 0 < δ ≤ 1, and β depending only on δ and the first eigenvalue λ 1 (M ) of Laplacian such that if M satisfies a (δ-SC) condition and λ 1 (M ) has a lower bound then H 1 (L 2 (M )) = 0. Excepting these two conditions, there is no more additional condition on the curvature.