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 $\overline{M}=I{\times }_{\lambda }M$. Based on the conjecture proposed by Ren–Wang in Ren and Wang (2020), which is valid for $k\ge n-2$, we derive curvature estimates for equation ${\sigma }_k\left(\kappa \right)=\psi (V,\nu \left(V\right))$ through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface $\Sigma$ satisfying the above equation by the degree theory under some sufficient conditions.