Families of short cycles on Riemannian surfaces

TL;DR

This paper constructs families of short cycles on Riemannian surfaces that represent non-trivial homology classes, with bounds on their length depending on genus and homology index, showing optimality up to a constant.

Contribution

It introduces a method to build families of cycles with controlled length representing homology classes on Riemannian surfaces, achieving near-optimal bounds.

Findings

Constructed families of cycles with length bounds depending on genus and homology index.

Proved the bounds are optimal up to a multiplicative constant.

Established a link between cycle length and topological complexity of the surface.

Abstract

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a non-trivial element of the k'th homology group of the space of cycles and such that the mass of each cycle is bounded above by $C \max\{\sqrt{k}, \sqrt{g}\} \sqrt{Area(M)}$. This result is optimal up to a multiplicative constant.