Filling surfaces with very few systoles

Olivier Mathieu

Filling surfaces with very few systoles

Metric Geometry 2026-06-27 v1 Differential Geometry

Authors: Olivier Mathieu

Abstract

In the paper we describe hyperbolic surfaces filled by their systoles, where the total number of systoles is in $O(\frac{g}{\ln \,g})$ , that is equivalent to the lower bound of Anderson, Parlier and Pittet \cite{APP}. Various papers \cite{SS}\cite{FB20}\cite{Sanki}\cite{ IM}\cite{ Mathieu} have investigated the same question, and the best previously known upper bounds where in $o(\frac{g}{{\sqrt{\ln \,g}}})$ . Surprizingly the present approach is, in our opinion, much simpler than the methods of earlier papers.

Cite

@article{arxiv.2606.28954,
  title  = {Filling surfaces with very few systoles},
  author = {Olivier Mathieu},
  journal= {arXiv preprint arXiv:2606.28954},
  year   = {2026}
}