Shape of filling-systole subspace in surface moduli space and critical points of systole function

Abstract

This paper studies the space $X_g\subset \mathcal{M}_g$ consisting of surfaces with filling systoles and its subset, critical points of the systole function. In the first part, we obtain a surface with Teichm\"uller distance $\frac{1}{5}\log\log g$ to $X_g$ and in the second and third part, prove that most points in $\mathcal{M}_g$ have Teichm\"uller distance $\frac{1}{5}\log\log g$ to $X_g$ and Weil-Petersson distance $0.6521(\sqrt{\log g}-\sqrt{7\log\log g})$ respectively.Therefore we prove that the radius-$r$ neighborhood of $X_g$ is not able to cover the thick part of $\mathcal M_g$ for any fixed $r>0$. In last two parts, we get critical points with small and large (comparable to diameter of thick part of $\mathcal M_g$) distance respectively.