k-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds

In this paper, we consider Weingarten curvature equations for k-convex hypersurfaces with n<2k in a warped product manifold M=I_M. Based on the conjecture proposed by Ren-Wang in Ren2, which is valid for k n-2, we derive curvature estimates for equation ₖ () = (V, (V) ) through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface satisfying the above equation by the degree theory under some sufficient conditions.