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

Abstract We derive global curvature estimates for closed, strictly star-shaped $$(n-2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> -convex hypersurfaces in warped product manifolds, which satisfy the prescribed $$(n-2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> -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.