Vanishing theorems on Hypersurfaces in S^n R

We discuss a complete noncompact hypersurface ⁿ in a product manifold S^n R (n 3). Suppose that the inner product of the unit normal to and t has a positive lower bound ₀, where t denotes the coordinate of the factor R of S^n R. We prove that there is no nontrivial L² harmonic 1-form if the total curvature or the length of the traceless of the second fundamental form is bounded from above by a constant depending only on n and ₀. These results are extensions of results on hypersurfaces in Hadamard manifolds and spheres. These results are also generalization of results on hypersurfaces in S^n R without minimality.