The supremum of conformally covariant eigenvalues in a conformal class

Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension >1

Minoration du spectre des variétés hyperboliques de dimension 3

20 pages, 1 figureInternational audienceLet $M$ 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 $\mu_1(M_i)\geq \frac c{d_i^2}$.Soit $M$ une variété hyperbolique compacte de dimension~3, de diamètre~$d$ et de volume $\leq V$. Si on note $\mu_i(M)$ la $i$-ième valeur propre du laplacien de Hodge-de~Rham agissant sur les 1-formes coexactes de $M$, on montre que $\mu_1(M)\geq \frac c{d^3e^{2kd}}$ et $\mu_{k+1}(M)\geq \frac c{d^2}$, où $c>0$ est une constante ne dépendant que de $V$, et $k$ est le nombre de composantes connexes de la partie mince de $M$. En outre, on montre que pour toute 3-variété hyperbolique $M_\infty$ de volume fini avec cusps, il existe une suite $M_i$ de remplissages compacts de $M_\infty$, de diamètre $d_i\to+\infty$ telle que et $\mu_1(M_i)\geq \frac c{d_i^2}$

Prescription du spectre de Steklov dans une classe conforme

On any compact manifold of dimension $n\geq3$ with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface with boundary, we show that the multiplicity of the $k$-th eigenvalue is bounded independently of the metric. On the disk, we give more precise results : the multiplicity of the first and second positive eigenvalues are at most 2 and 3 respectively. For the Steklov-Neumann problem on the disk, we prove that the multiplicity of the $k$-th positive eigenvalue is at most $k+1$.Comment: 27pages, in French, 1 figur

Prescription de la multiplicit\'e des valeurs propres du laplacien de Hodge-de Rham

On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum Hodge Laplacian acting on $p$-form for $1\leq p<\frac n2$. In particular, we prescribe the multiplicity of the first eigenvalues.Comment: 19 pages, in frenc