Bounding immersed curves

This paper addresses the immersion problem raised by H. Hopf and R. Thorn in Séminaire Bourbaki 1957–1958 (Exposé 157) as reported in V. Poénaru (1967/68). Let γ be a smooth closed connected curve on a surface M. The immersion problem is: when does γ extend to an immersion in M of a surface with boundary equal to γ? We study curves on closed orientable surfaces and give a solution in the case that the genus g ≠ 1.