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 $g + n \geq 5$, where $g$ is the genus of the surface and $n$ is the number of the boundary components.