Starshaped compact hypersurfaces in warped product manifolds I: prescribed curvature equations

We derive global curvature estimates for closed, strictly star-shaped $(n-2)$-convex hypersurfaces in warped product manifolds, which satisfy the prescribed $(n-2)$-curvature equation with a general right-hand side. The proof can be readily adapted to establish curvature estimates for semi-convex solutions to the general $k$-curvature equation. Furthermore, it can also be used to prove the same estimates for $k$-convex solutions to the prescribed curvature measure type equations.