Abstract
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 monotonically 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.
Citation
Erin Wolf Chambers. Gregory R. Chambers. Arnaud de Mesmay. Tim Ophelders. Regina Rotman. "Constructing monotone homotopies and sweepouts." J. Differential Geom. 119 (3) 383 - 401, November 2021. https://doi.org/10.4310/jdg/1635368350
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