Non-singular extensions of circle-valued Morse functions

In this paper, we consider the non-singular extension problem for circle-valued Morse functions on closed orientable surfaces. The problem asks, given a circle-valued Morse function $f\text{\hspace{0.17em}⁣}:M\to S^1$ on a closed orientable surface $M$, under what condition there exist a compact orientable 3-dimensional manifold $N$ with $\mathrm{\partial }N=M$ and a submersion $G\text{\hspace{0.17em}⁣}:N\to S^1$ such that $G{\mathrm{\mid }}_{\mathrm{\partial }N}=f$. We provide necessary and sufficient conditions for the existence of a non-singular extension of a circle-valued Morse function as the main theorem when a submersion on a collar neighborhood is given.