Rigidity of Immersed Submanifolds in a Hyperbolic Space

Let <TEX>Mⁿ</TEX>, <TEX>2≤n≤6</TEX> be a complete noncompact hypersurface immersed in <TEX>Hⁿ⁺¹</TEX>. We show that there exist two certain positive constants 0 < <TEX>δ≤1</TEX>, and <TEX>β</TEX> depending only on <TEX>δ</TEX> and the first eigenvalue <TEX>λ₁(M)</TEX> of Laplacian such that if M satisfies a (<TEX>δ</TEX>-SC) condition and <TEX>λ₁(M)</TEX> has a lower bound then <TEX>H¹(L²(M))=0</TEX>. Excepting these two conditions, there is no more additional condition on the curvature.