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arXiv · 1704.06175

Constructing monotone homotopies and sweepouts

Erin Wolf Chambers·Gregory R. Chambers·Arnaud de Mesmay·Tim Ophelders·Regina Rotman
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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 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.

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Erin Wolf Chambers, Gregory R. Chambers, Arnaud de Mesmay, Tim Ophelders, Regina Rotman. 2017-04-20. Constructing monotone homotopies and sweepouts. https://arxiv.org/abs/1704.06175

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