Families of short cycles on Riemannian surfaces

Abstract

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ 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 $Cmax\hspace{0.17em}\{\sqrt{k},\sqrt{g}\}\sqrt{Area\left(M\right)}$. This result is optimal up to a multiplicative constant.