Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\Delta $ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda_i(q)$ the $i$-th eigenvalue of the Schrödinger type operator $-\Delta + q$ acting on functions with Dirichlet or Neumann boundary conditions in case $\partial M \neq \emp...
On montre l'équivalence entre l'hyperbolicité au sens de Gromov de la géométrie de Hilbert d'un domaine convexe du plan et la non nullité du bas du spectre de ce domaine
In this paper, we investigate the minimality of the map $\frac{x}{\|x\|}$ from the euclidean unit ball $\mathbf{B}^n$ to its boundary $\mathbb{S}^{n-1}$ for weighted energy functionals of the type $E_{p,f}= \int_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$, where $f$ is a non-negative function. We prove that in each of the two following cases:\\ i) $p=1$ a...
International audience
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of t...
National audience
National audience
Let G be a geometrically finite tree lattice. We prove a Khintchine-Sullivan type theorem for the Hausdorff measure of the points at infinity of the tree that are well approximated by the parabolic fixed points of G. Using Bruhat-Tits trees, an application is given for the Diophantine approximation of formal Laurent series in the variable 1/X over ...
metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the Dirichlet Laplacian. Using homogenization techniques we describe the asymptotic behavior of the spectrum when the radius of these balls goes to infinity. This inv...
Let $M$ be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any $(p_1,v_1)$ and $(p_2,v_2)$ in $TM$, is it possible to connect $p_1$ to $p_2$ by a curve $\gamma$ in $M$ with arbitrary small geodesic curvature such that, for $i=1,2$, $\dot \gamma$ is equal to $v_i$ at $p_i$? In this paper, we bring a...
