Related papers: Hyperbolic Eisenstein series for geometrically fin…
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…
We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as…
In the former part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical…
Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = \Gamma \backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $\gamma$ is parabolic, there…
We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichm\"uller theory than arbitrary non-compact surfaces. We show that the…
We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of…
We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable…
On convex co-compact hyperbolic surfaces with Hausdorff dimension of the limit set less than 1/2, we investigate high energy behaviour of Eisenstein Series. Eisenstein Series are non-L^2 eigenfunctions of the hyperbolic Laplacian which…
A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…
A new proof is given of Quantum Ergodicity for Eisenstein Series for cusped hyperbolic surfaces. This result is also extended to higher dimensional examples, with variable curvature.
We consider a magnetic Laplacian on a geometrically finite hyperbolic surface, when the corresponding magnetic field is infinite at the boundary at infinity. We prove that the counting function of the eigenvalues has a particular asymptotic…
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…
We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic…
In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $\pi_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we…
In this paper we review some author's results about Weingarten surfaces in Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property.…
We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds with universal cover a bounded domain is a finite set.
We study the sup-norm and mean-square-norm problems for Eisenstein series on certain arithmetic hyperbolic orbifolds, producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta…
We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}:…
For geometrically finite hyperbolic manifolds $\Gamma\backslash H^{n+1}$, we prove the meromorphic extension of the resolvent of Laplacian, Poincar\'e series, Einsenstein series and scattering operator to the whole complex plane. We also…
We present and prove a topological characterization of geodesic laminations on hyperbolic surfaces of finite type.