Vanishing theorems on Hypersurfaces in Sⁿ × R

Abstract

We discuss a complete noncompact hypersurface Σⁿ in a product manifold Sⁿ × R (n ≥ 3). Suppose that the inner product of the unit normal to Σ and ∂/∂t has a positive lower bound δ₀, where t denotes the coordinate of the factor R of Sⁿ × R. We prove that there is no nontrivial L² harmonic 1-form if the total curvature or the length of the traceless Φ of the second fundamental form is bounded from above by a constant depending only on n and δ₀. These results are extensions of results on hypersurfaces in Hadamard manifolds and spheres. These results are also generalization of results on hypersurfaces in Sⁿ × R without minimality.