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Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than , we construct a second sweepout composed of curves of length less than which are either constant curves or simple curves.
This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of [CL] concerning converting homotopies to isotopies in an effective way.
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Gregory R. Chambers, Yevgeny Liokumovich, Optimal sweepouts of a Riemannian 2-sphere. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1361–1377DOI 10.4171/JEMS/863