标题： 圆值莫尔斯函数的非奇异扩张 显示英文标题

标题： Non-singular extensions of circle-valued Morse functions

摘要： 本文中，我们考虑闭可定向曲面上圆值莫尔斯函数的非奇异扩展问题。 该问题要求，给定闭可定向曲面$M$上的圆值莫尔斯函数$f\colon M\to S^{1}$，在什么条件下存在一个紧致可定向三维流形$N$且$\partial N = M$，以及一个浸没$G\colon N \to S^{1}$，使得$G|_{\partial N}=f$。 当给出一个邻域上的浸没时，我们提供了圆值莫尔斯函数非奇异扩展存在的必要且充分条件作为主要定理。 显示英文摘要

摘要： 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 $f\colon M\to S^{1}$ on a closed orientable surface $M$, under what condition there exist a compact orientable 3-dimensional manifold $N$ with $\partial N = M$ and a submersion $G\colon N \to S^{1}$ such that $G|_{\partial N}=f$. 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.