Superinjective simplicial maps of the complexes of curves on nonorientable surfaces

We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if (g, n) in {(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)} or g + n ≥ 5, where g is the genus of the surface and n is the number of the boundary components.