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


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.


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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.


Received: 26 August 2017; Published: November 2021
First available in Project Euclid: 1 November 2021

Digital Object Identifier: 10.4310/jdg/1635368350

Rights: Copyright © 2021 Lehigh University


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Vol.119 • No. 3 • November 2021
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