Minoration du spectre des vari\'et\'es hyperboliques de dimension 3

TL;DR

This paper establishes lower bounds for the eigenvalues of the Hodge Laplacian on compact hyperbolic 3-manifolds, relating spectral properties to geometric features like diameter and volume, and explores limits for manifolds with cusps.

Contribution

It provides new explicit lower bounds for eigenvalues of the Hodge Laplacian on hyperbolic 3-manifolds, connecting spectral data with geometric parameters and analyzing sequences of manifolds with cusps.

Findings

Lower bounds depend on diameter and volume.

Eigenvalues grow at least as fast as inverse polynomial of diameter.

Sequences of manifolds with cusps have eigenvalues bounded below by inverse quadratic in diameter.

Abstract

Let $M$ be a compact hyperbolic 3-manifold of diameter $d$ and volume $\leq V$. If $\mu_i(M)$ denotes the $i$-th egenvalue of the Hodge laplacian acting on coexact 1-forms of $M$, we prove that $\mu_1(M)\geq \frac c{d^3e^{2kd}}$ and $\mu_{k+1}(M)\geq \frac c{d^2}$, where $c>0$ depends only on $V$, and $k$ is the number of connected component of the thin part of $M$. Moreover, we prove that for any finite volume hyperbolic 3-manifold $M_\infty$ with cusps, there is a sequence $M_i$ of compact fillings of $M_\infty$ of diameter $d_i\to+\infty$ such that $\mu_1(M_i)\geq \frac c{d_i^2}$.