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arXiv · 1312.3584

Hypersurfaces with constant curvature quotients in warped product manifolds

Jie Wu·Chao Xia
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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 $\frac{σ_{2k+1}}{σ_{2k}}$, where $σ_k$ are the usual mean curvatures. We prove several rigidity and Bernstein type results for compact or non-compact hypersurfaces corresponding to such quotients.

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Jie Wu, Chao Xia. 2013-12-12. Hypersurfaces with constant curvature quotients in warped product manifolds. https://doi.org/10.2140/pjm.2015.274.355

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