# The maximum number of systoles for genus two Riemann surfaces with abelian differentials

## TL;DR

This paper establishes bounds on the number of systoles on genus two Riemann surfaces with abelian differentials, showing the maximum is ten and identifying the unique surface achieving this, with extensions to higher genus.

## Contribution

It provides the first explicit bounds on systoles for genus two surfaces with abelian differentials and characterizes the extremal cases, extending to higher genus with specific bounds.

## Key findings

- Maximum of 10 systoles for genus two surfaces with abelian differentials.
- Unique translation surface up to homothety attains this maximum.
- Optimal bounds of 6g-3 and 6g-5 for general and hyperelliptic cases, respectively.

## Abstract

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,\omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.