An upperbound for the number of critical points of the systole function on the surface moduli space

TL;DR

This paper establishes an upper bound on the number of critical points of the systole function in the surface moduli space, including those with systole below a certain threshold, advancing understanding of the function's critical structure.

Contribution

It provides the first known upper bounds for the critical points of the systole function on moduli space, including those with small systole values.

Findings

Upper bound for total critical points of the systole function

Upper bound for critical points with systole below a constant

Insights into the distribution of critical points in moduli space

Abstract

We obtain an upper bound for the number of critical points of the systole function on $\mathcal{M}_g$. Besides, we obtain an upper bound for the number of those critical points whose systole is smaller than a constant.