The shape of the filling-systole subspace in surface moduli space and critical points of the systole function

Gao, Yue

doi:10.2140/agt.2024.24.2011

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The shape of the filling-systole subspace in surface moduli space and critical points of the systole function

Yue Gao

Algebraic & Geometric Topology 24 (2024) 2011–2038

DOI: 10.2140/agt.2024.24.2011

Abstract

We study the space $X_{g} \subset ℳ_{g}$ consisting of surfaces with filling systoles and its subset, critical points of the systole function. In the first part we obtain a surface with Teichmüller distance $\frac{1}{5} \log \log g$ to $X_{g}$ , and in the second and third parts prove that most points in $ℳ_{g}$ have Teichmüller distance $\frac{1}{5} \log \log g$ and Weil–Petersson distance $0.6521 (\sqrt{\log g} - \sqrt{7 \log \log g})$ to $X_{g}$ . So the radius- $r$ neighborhood of $X_{g}$ cannot cover the thick part of $ℳ_{g}$ for any fixed $r > 0$ . In the last two parts, we get critical points with small and large (comparable to the diameter of the thick part of $ℳ_{g}$ ) distances.

Keywords

filling systole subspace, sparseness, random surfaces, Weil–Petersson metric, critical points of the systole function

Mathematical Subject Classification

Primary: 30F45, 30F60, 57K20

References

Publication

Received: 7 February 2022

Revised: 17 January 2023

Accepted: 27 April 2023

Published: 16 July 2024

Authors

Yue Gao
School of Mathematics and Statistics Anhui Normal University Wuhu China

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Volume 24, issue 4 (2024)