Minimal hypersurfaces in R^n \times S^m

We classify minimal hypersurfaces in $R^n\times S^m$, $n,m\ge 2$, which are invariant by the canonical action of $O(n)\times O(m)$. We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvature hypersurfaces are all unstable.