Small eigenvalues of random 3-manifolds

TL;DR

This paper demonstrates that in random closed 3-manifolds with a fixed Heegaard genus, the smallest positive eigenvalue diminishes at a rate inversely proportional to the square of the manifold's volume, highlighting a relationship between spectral properties and geometric complexity.

Contribution

It establishes a bound on the smallest positive eigenvalue for random 3-manifolds of fixed genus, linking spectral and geometric properties in a probabilistic setting.

Findings

Smallest eigenvalue is at most c(g)/vol(M)^2 for random 3-manifolds.

The bound depends on the genus g of the manifold.

Provides a probabilistic relationship between eigenvalues and volume.

Abstract

We show that for every $g\geq 2$ there exists a number $c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{\rm vol}(M)^2$.