Let be a closed Riemannian surface of genus . We construct a family of 1-cycles on that represents a nontrivial element of the kth homology group of the space of cycles and such that the mass of each cycle is bounded above by . This result is optimal up to a multiplicative constant.
Yevgeny Liokumovich. "Families of short cycles on Riemannian surfaces." Duke Math. J. 165 (7) 1363 - 1379, 15 May 2016. https://doi.org/10.1215/00127094-3450208