[Paper Review] Towards optimal spectral gaps in large genus

Michael Lipnowski, Alex Wright|arXiv (Cornell University)|Mar 12, 2021

Geometric and Algebraic Topology9 citations

TL;DR

This paper establishes that, in the large genus limit, Weil-Petersson typical hyperbolic surfaces have their first non-zero Laplace eigenvalue $\lambda_1$ bounded away from zero, with probability approaching one, $\lambda_1$ exceeds $\frac{3}{16} - \varepsilon$ for any $\varepsilon > 0$. The proof combines Mirzakhani's volume integration techniques with the Selberg trace formula, analyzing the distribution of short geodesics and controlling error terms via refined volume bounds and local Weyl laws.

ABSTRACT

We show that the Weil-Petersson probability that a random surface has first eigenvalue of the Laplacian less than $3/16-ε$ goes to zero as the genus goes to infinity.

Motivation & Objective

• To understand the typical behavior of the first non-zero Laplace eigenvalue $\lambda_1$ on hyperbolic surfaces of large genus.
• To determine whether $\lambda_1$ is typically close to the theoretical upper bound of $\frac{1}{4}$ in high genus.
• To extend previous results on geodesic counting and spectral gaps by allowing length scales $L$ growing with genus.
• To provide improved error terms in limit multiplicity laws and spectral gap estimates via averaging the Selberg trace formula.

Proposed method

• Averaging the Selberg trace formula over the Weil-Petersson measure on moduli space $\mathcal{M}_g$ to relate spectral data to geodesic counting.
• Using Mirzakhani’s volume integration formula to estimate the number of simple, non-separating geodesics of length at most $L$.
• Establishing new bounds on the number of subsurfaces filled by non-simple geodesics to control error terms in the trace formula.
• Applying a local Weyl law to bound the number of spectral parameters in intervals, incorporating systole and geodesic count dependencies.
• Introducing a refined volume polynomial analysis to control contributions from multi-geodesics and subsurfaces of fixed area.
• Using a cutoff function $F$ supported on $[0, D\log g]$ to control the trace formula's contribution from short geodesics.

Experimental results

Research questions

• RQ1 What is the typical value of the first non-zero Laplace eigenvalue $\lambda_1$ on a random hyperbolic surface of large genus?
• RQ2 Can the spectral gap be improved beyond $\frac{3}{16}$ using trace formula averaging and geodesic counting?
• RQ3 How do the proportions of simple versus non-simple geodesics change as genus increases and length scales grow?
• RQ4 What is the role of the systole and number of short geodesics in bounding spectral parameters?
• RQ5 Can error terms in the Selberg trace formula be controlled uniformly across large genus families?

Key findings

• For any $\varepsilon > 0$, the Weil-Petersson probability that $\lambda_1 < \frac{3}{16} - \varepsilon$ tends to zero as genus $g \to \infty$.
• The average of $F_{\text{all}}$ over a thickened moduli space $\mathcal{M}_g'$ is bounded by $I_F + O(g^{-1+\kappa} I_{\widetilde{F}})$, where $I_F = \int_0^\infty F(\ell) \ell \frac{\sinh(\ell/2)^2}{(\ell/2)^2} d\ell$.
• The $\mu$-probability that $\lambda_1 \leq \frac{1}{4} - b^2$ is at most $O\left(g^{1 - 4b(1 - \kappa/2) + o(1)}\right)$, showing decay for $b > 0$.
• The number of multi-geodesics with $k$ components and total length $\leq L$ is bounded by $O(e^L L^{2k} g^{-a})$ for subsurfaces of area $2\pi a$.
• The number of primitive closed geodesics of length $\leq \mu_0$ is bounded by $3g - 3$ for sufficiently small $\mu_0$, and the number of exceptional spectral parameters is at most $2g - 3$.
• A local Weyl law bounds the number of real spectral parameters in $[-1+t, 1+t]$ by $O(g \cdot \overline{\log}(1/\mathrm{sys}(X)))$, with constants depending on $\mu_0$.