Biconservative and Biharmonic Hypersurfaces in Product Spaces

Abstract

In this paper we investigate biconservative and biharmonic hypersurfaces in product spaces $L^m(\epsilon )\times \mathbb{R}$, where $L^m(\epsilon )$ denotes the space form ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ of constant sectional curvature $\epsilon =1$ or $\epsilon =-1$ respectively. We obtain their complete description in the case of isoparametric hypersurfaces and hypersurfaces of constant angle. Moreover, we prove some classification results for biconservative and biharmonic hypersurfaces for which the gradient of the mean curvature function is a principal direction. In this context we obtain new examples of complete, non-cylindrical, non-totally umbilical biconservative hypersurfaces in ${\mathbb{H}}^m\times \mathbb{R}$. Then we prove a rigidity theorem related to these examples. Finally, we prove a global non-existence theorem for complete proper biharmonic hypersurfaces in ${\mathbb{H}}^m\times \mathbb{R}$ such that the square of the mean curvature function is bounded and the Ricci curvature is bounded from below.