Constructing monotone homotopies and sweepouts (Q2073264)

The article investigates the conditions under which homotopies can be improved, without increasing the lengths of curves, to monotone homotopies, that is homotopies consisting of curves which are simple or constant, and in which curves are pairwise disjoint. To this end, the authors first prove that, if the boundary of a Riemannian disc can be contracted through curves of length less than \(L\), then it can also be contracted monotonically through curves of length less than \(L\). This results proves a conjecture of \textit{G. R. Chambers} and \textit{R. Rotman} [J. Topol. Anal. 10, No. 2, 323--354 (2018; Zbl 1395.53035)]. They furthermore demonstrate that 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 \(L\). The above results have numerous applications to metric geometry and applied topology. In particular, the first one improves known estimates of the lengths of the shortest geodesics between pairs of points on Riemannian \(2\)-spheres, while the second one has applications, inter alia, in the study of minimal surfaces.