This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.

中文翻译：

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构造单调同伦和扫除

本文研究何时可以在不增加曲线长度的情况下将同伦转换为单调同伦。单调同伦是由简单的或恒定的曲线组成的，其中曲线是成对不相交的。我们证明，如果黎曼圆盘的边界可以通过长度小于 $L$ 的曲线收缩，那么它也可以通过长度小于 $L$ 的曲线单调收缩。这证明了钱伯斯和罗特曼的猜想。此外，黎曼$2$-球体通过长度小于$L$ 的曲线的任何扫掠都可以替换为通过长度小于$L$ 的曲线的单调扫掠。还讨论了这些结果的应用。