Closed Weingarten hypersurfaces in warped product manifolds

Details Closed Weingarten hypersurfaces in warped product manifolds Closed Weingarten hypersurfaces in warped product manifolds

2008-10-18

arxiv.org/abs/0810.3306v1

Mathematics

Differential Geometry

Paper accepted to publication in Indiana University Mathematics Journal

Scientific paper

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function $\psi$ in $\bar M$, we find a closed hypersurface $\Sigma$ which is solution of an equation of the form $F(B)=\psi$, where $B$ is the second fundamental form of $\Sigma$ and $F$ is a function satisfying certain structural properties. As examples, we may exhibit examples of hypersurfaces with prescribed higher order mean curvature.

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