Constructing monotone homotopies and sweepouts

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