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We compute the Laplacian spectra of singular area-minimising hypersurfaces in the hyperbolic space with prescribed asymptotic data. We also obtain similar results in higher codimension, and explore related extremal properties of the bottom…
The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.
We review some results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian.
We study the spectrum of the Laplacian on two models of random hyperbolic 3-orbifolds, related to the Apollonian group and the super Apollonian group. We determine explicit spectral gaps for these random orbifolds. Moreover, we use our…
We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the…
We compute numerically eigenvalues and eigenfunctions of the Laplacian in a three-dimensional hyperbolic space. Applying the results to cosmology, we demonstrate that the methods learned in quantum chaos can be used in other fields of…
The spectral geometry of negatively curved manifolds has received more attention than its positive curvature counterpart. In this paper we will survey a variety of spectral geometry results that are known to hold in the context of…
In this paper, we establish a sharp lower bound for the spectrum of the Hodge Laplacian on K\"ahler hyperbolic manifolds. This bound is expressed explicitly in terms of the supremum norm of the 1-form associated with the K\"ahler hyperbolic…
Varying physical constant cosmologies were claimed to solve standard cosmological problems such as the horizon, the flatness and the $\Lambda$-problem. In this paper, we suggest yet another possible application of these theories: solving…
We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.
We study two quantifications of being a homology sphere for hyperbolic 3-manifolds, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. We first…
We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectrums of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the…
Invited contribution to the Encyclopedia of Mathematical Physics (2nd edition), providing an overview over some main ideas and results in quantum cosmology. Key points: Canonical quantisation of homogeneous, isotropic cosmology; discussion…
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the…
We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of…
For a geometrically finite hyperbolic surface of infinite volume we write down the spectral decomposition for the Laplacian on 1-forms, generalize the Kudla and Millson's construction of hyperbolic Eisenstein series and other related…
A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian…
For certain compactly supported metric and/or potential perturbations of the Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the…
The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions…
We consider different aspects of the problem of cosmological singularity such as the BKL oscillatory approach to the singularity, the new features of the cosmological dynamics in the neighbourhood of the singularity in multidimensional and…