Related papers: Explicit spectral gap for Schottky subgroups of $\…

We show that for a very general and natural class of curvature functions, the problem of finding a complete strictly convex hypersurface satisfying f({\kappa}) = {\sigma} over (0,1) with a prescribed asymptotic boundary {\Gamma} at infinity…

On a fairly general class of Riemannian manifolds M, we prove lower estimates in terms of the Ricci curvature for the spectral bound (when M has infinite volume) and for the spectral gap (when M has finite volume) for the Laplace-Beltrami…

We give a lower bound for the bottom of the $L^2$ differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent…

We show that for any $\epsilon>0$, $\alpha\in[0,\frac{1}{2})$, as $g\to\infty$ a generic finite-area genus g hyperbolic surface with $n=O\left(g^{\alpha}\right)$ cusps, sampled with probability arising from the Weil-Petersson metric on…

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have…

We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $\Delta_L$ contains the ray $[{1/4},+\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional…

Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this…

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as…

Let F be a square integrable Maass form on the Siegel upper half space of rank 2 for the Siegel modular group Sp(4, Z) with Laplace eigenvalue lambda. If, in addition, F is a joint eigenfunction of the Hecke algebra, we show a power-saving…

Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for $PSL_2(O)$ in the upper-half space model of hyperbolic space, where $O$ is an imaginary quadratic ring of integers with…

Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2,R). We consider the closed orbit generated by a specific covering of degree 7 of the…

We extend the Otal-Rosas bound on the number of small eigenvalues of the Laplacian on a hyperbolic surface to the small eigenvalues of pseudo-Laplacians. In the process, we extend the work of Colin de Verdi\`ere on the spectral theory of…

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this…

We prove the existence of a pair $(\Sigma ,\, \Gamma)$, where $\Sigma$ is a compact Riemann surface with $\text{genus}(\Sigma)\, \geq\, 2$, and $\Gamma\, \subset\, {\mathrm SL}(2, \mathbb C)$ is a cocompact lattice, such that there is a…

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$\chi$, and let $\lambda_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, $\lambda_{-\chi}>\frac14$.…

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1-->…

In this paper we study the operator $H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}(\mathbb{R}^{2})$, where $\Gamma$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the asymptotic form of the band spectrum of $H_{\beta}$…

Let $\Gamma$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $\Gamma$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $\Gamma$ is…

Let $\Lambda$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=\Lambda \backslash\H$ has infinitely many resonances in…

Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation…