Let u be a harmonic function defined on a domain of a space form.We establish a constant rank theorem for the second fundamental form of the convex level sets of u.Applying the deformation process, we prove that the level sets of u on a convex ring are strictly convex.Moreover we give a lower bound for the Gaussian curvature of the convex level sets in terms of the Gaussian curvature of the boundary and the the gradient of u on the boundary.