Vanishing theorems on Hypersurfaces in $\mathbf{S}^{n} \times \mathbf{R}$

Abstract

We discuss a complete noncompact hypersurface $\Sigma^n$ in a product manifold $\mathbf{S}^{n} \times \mathbf{R} (n \geq 3)$. Suppose that the inner product of the unit normal to $\Sigma$ and $\frac{\partial}{\partial t}$ has a positive lower bound $\delta_0$, where $t$ denotes the coordinate of the factor $\mathbf{R}$ of $\mathbf{S}^{n} \times \mathbf{R}$. We prove that there is no nontrivial $L^2$ harmonic 1-form if the total curvature or the length of the traceless $\Phi$ of the second fundamental form is bounded from above by a constant depending only on $n$ and $\delta_0$. These results are extensions of results on hypersurfaces in Hadamard manifolds and spheres. These results are also generalization of results on hypersurfaces in $\mathbf{S}^{n} \times \mathbf{R}$ without minimality.