Closed Weingarten hypersurfaces in warped product manifolds

Abstract

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.

Paper accepted to publication in Indiana University Mathematics Journal