Related papers: Automorphic Spectra and the Conformal Bootstrap

Let $\mathcal{M}=\Gamma\backslash\mathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in…

In this article, we generalize the set of manifolds over which the $L^p$-spectrum of the Laplacian on $k$-forms depends on $p$. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic…

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or…

Properly modeling and predicting the scattering response of a metasurface is a particularly challenging task. This has been shown to be especially difficult if the metasurface supports both local and nonlocal interactions, in the form of…

On compact Riemann surfaces, the Laplacian has a discrete, non-negative spectrum of eigenvalues of finite multiplicity. The spectrum is intrinsically linked to the geometry of the surface. In this work, we consider surfaces of constant…

We study extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of compact genus two Riemann surfaces. By a combination of analytical and numerical methods we identify four non-degenerate critical…

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces, we introduce a dynamical invariant, denoted by N, for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed…

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only…

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real-valued potentials. For $L^1$-potentials, we obtain optimal…

This work investigates upper bounds for the spectrum of the Steklov-type operator on Riemannian manifolds with boundary. We extend the Fraser-Schoen estimate for the first positive Steklov eigenvalue to higher Steklov eigenvalues, in terms…

We show a uniform spectral gap of stable commutator length for all compact hyperbolic $2$-orbifolds relative to the peripheral subgroups. Except for the case of a sphere with three cone points, we have an explicit uniform gap $1/36$. These…

We adapt modulus of continuity estimates to the study of spectra of combinatorial graph Laplacians, as well as the Dirichlet spectra of certain weighted Laplacians. The latter case is equivalent to stoquastic Hamiltonians and is of current…

Let $\mathcal{M}_{g}$ be the moduli space of compact connected hyperbolic surfaces of genus $g\geq2$, and ${\mathcal B}_g \subset {\mathcal M}_{g} $ its branch locus. Let $\widehat{{\mathcal{M}}_{g}}$ be the Deligne-Mumford compactification…

We present a few subjects from physics that have one in common: the spectral resolution of the Laplacian.

We investigate the spectrum of the Laplacian on complete, non-compact manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1 \in…

We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…

For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing…

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the…

In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial…