Minoration du spectre des variétés hyperboliques de dimension 3

Abstract

Let $M$ be a compact hyperbolic 3-manifold of diameter $d$ and volume $\le V$. If ${\mu }_i(M)$ denotes the $i$-th egenvalue of the Hodge laplacian acting on coexact 1-forms of $M$, we prove that ${\mu }_1(M)\ge \frac{c}{d^3{e}^{2kd}}$ and ${\mu }_{k+1}(M)\ge \frac{c}{d^2}$, 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 ${\mu }_1(M_i)\ge \frac{c}{d_i^2}$.

arXiv comment20 pages, 1 figure, in french