Journal of the European Mathematical Society

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Volume 21, Issue 5, 2019, pp. 1361–1377
DOI: 10.4171/JEMS/863

Published online: 2019-01-11

Optimal sweepouts of a Riemannian 2-sphere

Gregory R. Chambers[1] and Yevgeny Liokumovich[2]

(1) Rice University, Houston, USA
(2) Institute for Advanced Study, Princeton, USA

Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than $L$, we construct a second sweepout composed of curves of length less than $L$ 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.

Keywords: Homotopies of curves, sweepouts of Riemannian spheres, embedded geodesics, minmax constructions

Chambers Gregory, Liokumovich Yevgeny: Optimal sweepouts of a Riemannian 2-sphere. J. Eur. Math. Soc. 21 (2019), 1361-1377. doi: 10.4171/JEMS/863