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

TL;DR

This paper investigates the geometric structure of the moduli space of surfaces with filling systoles, analyzing the distances to critical points of the systole function and revealing limitations of neighborhoods around these points in covering the space.

Contribution

It provides new bounds on the distances in moduli space related to filling systoles and critical points, demonstrating the complexity of the space's geometry.

Findings

Most points in moduli space are at a logarithmic distance from the filling-systole subspace.

Neighborhoods around the filling-systole subspace cannot cover the entire thick part of the moduli space.

Critical points of the systole function exist at both small and large distances, comparable to the diameter of the thick part.

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.