Closed Weingarten hypersurfaces in warped product manifolds

Abstract

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