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 spaces, satisfying the prescribed (n-2)-curvature equation with a general right-hand side. The proof is inspired by the recent breakthrough of Guan-Ren-Wang CPAM 2015 and it can be readily adapted to establish curvature estimates for semi-convex and (k+1)-convex solutions to the general k-curvature equation. Furthermore, it can also be used to prove the same estimates for prescribed curvature measure type equations.