Hypersurfaces with constant curvature quotients in warped product manifolds

Jie Wu and Chao Xia

Vol. 274 (2015), No. 2, 355–371

DOI: 10.2140/pjm.2015.274.355

Abstract

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

Keywords

constant mean curvature, rigidity, warped product manifold, Gauss–Bonnet curvature

Mathematical Subject Classification 2010

Primary: 53C24

Secondary: 52A20, 53C40

Milestones

Received: 11 December 2013

Revised: 30 June 2014

Accepted: 11 September 2014

Published: 1 April 2015