Small eigenvalues of hyperbolic surfaces with many cusps

TL;DR

This paper establishes topological lower bounds on the number of small Laplacian eigenvalues for hyperbolic surfaces with many cusps, revealing how geometric and topological features influence spectral properties.

Contribution

It provides new bounds on small eigenvalues of hyperbolic surfaces with cusps, linking eigenvalue counts to genus and cusp number, and improves bounds under additional geometric constraints.

Findings

Existence of constants a, b > 0 relating eigenvalues to genus and cusps

Lower bounds on small eigenvalues depend on the ratio of genus and cusp count

Improved bounds are possible with constraints on short geodesics

Abstract

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at least $b\frac{2g+n-2}{\log\left(2g+n-2\right)}$ Laplacian eigenvalues below $\frac{1}{4}$. We also show that, under certain additional constraints on the lengths of short geodesics, the lower bound can be improved to $b\left(2g+n-2\right)$ with the weaker condition $(g+1)<an$.