Hypersurfaces with constant curvature quotients in warped product manifolds

Abstract

In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients $\frac{{\mathcal{H}}_{2k+1}}{{\mathcal{H}}_{2k}}$ in the warped product manifolds. Here ${\mathcal{H}}_{2k}$ is the $k$-th Gauss-Bonnet curvature and ${\mathcal{H}}_{2k+1}$ arises from the first variation of the total integration of ${\mathcal{H}}_{2k}$. Hence the quotients considered here are in general different from $\frac{{\sigma }_{2k+1}}{{\sigma }_{2k}}$, where ${\sigma }_k$ are the usual mean curvatures. We prove several rigidity and Bernstein type results for compact or non-compact hypersurfaces corresponding to such quotients.

arXiv comment15 pages

References in corpus (2)

• Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes
• Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson-Walker spacetimes