Spectral gaps on thick part of moduli spaces

TL;DR

This work establishes that for any fixed $k\ge 1$ and any thickness parameter $\epsilon>0$, the maximal spectral gap $\lambda_k-\lambda_{k-1}$ on the $\epsilon$-thick part of the genus-$g$ moduli space tends to $\tfrac{1}{4}$ as $g\to\infty$. The authors blend a three-pronged strategy: (i) compare spectra via a funnel-core reduction and use random-covering constructions to retain large $\bar{\lambda}_1$ while controlling systoles, (ii) prove uniform stability of the first Neumann eigenvalue under small boundary deformations, and (iii) glue multiple thick pieces obtained from random-surface inputs to realize closed surfaces in ${\mathcal M}_g^{\geq\epsilon}$ with a large $k$-th spectral gap, leveraging the mini-max principle. The main result extends the thick-part regime of spectral-gap phenomena beyond previous boundary-based compactifications, showing $\lim_{g\to\infty}\max_{X_g\in{\mathcal M}_g^{\geq\epsilon}}(\lambda_k-\lambda_{k-1})=\tfrac{1}{4}$ for all fixed $k$ and $\epsilon>0$. This advances understanding of spectral geometry on moduli spaces and highlights the interplay between hyperbolic geometry, random surfaces, and spectral theory, with potential arithmetic implications for special families of surfaces.

Abstract

In this paper, we study spectral gaps of closed hyperbolic surfaces for large genus. We show that for any fixed $k\geq 1$, as the genus goes to infinity, the maximum of $λ_k-λ_{k-1}$ over any thick part of the moduli space of closed Riemann surfaces approaches the limit $\frac{1}{4}$.

Figures (11)

• Figure 1: An illustration from $X$ to $Y$.
• Figure 2: The construction of desired closed hyperbolic surfaces of large genus in Theorem \ref{['main']} when $k=4$.
• Figure 3: Cusp, collar and funnel
• Figure 4: An illustration for the metric $ds_{X^\delta}^2$ on $\mathop{\mathrm{Cusp}}\nolimits_i(l)$.
• Figure 5: The location of $\gamma_i$ in the proof of Lemma \ref{['lem:comparison_geometry_lem4']}.

Theorems & Definitions (124)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark
• Lemma 2.3