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Due to the isotropy of $d$-dimensional hyperbolic space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hyperboloid model of hyperbolic geometry…
We study the spectrum of the Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities and its dependence on the generalized Dehn surgery coefficients.
Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a…
In this paper, we show the spectral convergence result of $\overline{\partial}$-Laplacians when $(X,\omega)$ is a compact toric symplectic manifold equipped with the natural prequantum line bundle $L$. We consider a family $\{ J_s\}_s$ of…
The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement $\Omega := \{(z,w) \in \widehat{\mathbb{C}}^2 \colon z \cdot w \neq 1\}$ of the "complexified unit circle" $\{(z,w) \in…
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…
In this paper we present a spectral decomposition of solutions to relativistic wave equations described on horizon penetrating hyperboloidal slices within a given Schwarzschild-black-hole background. The wave equa- tion in question is…
For a class of manifolds that includes quotients of real hyperbolic space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian coincide, with multiplicities,…
In this article we establish the radial symmetry of positive solutions of a p- Laplace equation in the Hyperbolic space, which is the Euler Lagrange equation of the p- Poincare Sobolev inequality in the Hyperbolic space. We will also…
We study relationships between asymptotic geometry of submanifolds in the hyperbolic space and their regularity properties near the ideal boundary, revisiting some of the related results in the literature. In particular, we discuss…
We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.
We derive lower bounds for the essential spectrum of the Hodge-Laplacian on geometrically finite orbifolds and their suborbifolds.
Astrophysical bounds on the cosmological constant are examined for spherically symmetric bodies. Similar limits emerge from hydrostatical and gravitational equilibrium and the validity of the Newtonian limit. It is argued that the bound…
We consider an inverse problem associated with $n$-dimensional asymptotically hyperbolic orbifolds $(n \geq 2)$ having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a…
This article presents some methods to control the bottom of the spectrum of the Laplacian $\lambda_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $\lambda_0$ of a geometrically finite surface in terms of the…
A Lindblad master equation for a harmonic oscillator, which describes the dynamics of an open system, is formally solved. The solution yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are…
The magnetic Laplacian on hyperbolic surfaces provides a rich analytic framework in which a variety of quantum phenomena emerge. The present note, written for the \emph{Proceedings of the Journ\'ees EDP 2025}, is a concise overview of the…
We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on M)/Ker(d)?
We recently proposed a chameleonic solution to the cosmological constant problem - Phys. Rev. D82 (2010) 044006. One of the results of that paper is a non-equivalence of different conformal frames at the quantum level. In this letter we…
The nature of space-time at high energy is an open question and the link between extra-dimensional theories with the physics of the Standard Model can not be established in a unique way. The compactification path is not unique and…