In this article, we consider complete, n-dimensional, Riemannian manifolds of finite volume. We assume that the Ricci curvature is bounded from below and normalized to have the lower bound given by Ric M ≥ −(n − 1).

For a compact Riemannian manifold, Weyl’s law describes the asymptotic behavior of the counting function of the eigenvalues of the associated Laplace operator. In this paper we discuss Weyl’s law in the context of automorphic forms. The underlying manifolds are locally symmetric spaces of finite volume. In the non-compact case Weyl’s law is closely related to the problem of existence of cusp fo...

We give bounds on finite volume expectations for a set of boundary conditions containing the support of any tempered Gibbs state and prove a theorem connecting the behavior of Gibbs states to the differentiability of the pressure for continuum statistical mechanical systems with long range superstable potentials. Convergence of grandcanonical Gibbs states is also studied. This is an extended ve...

Let IH be the n-dimensional hyperbolic space and let P be a simple polytope in IH. P is called an ideal polytope if all vertices of P belong to the boundary of IH. P is called a Coxeter polytope if all dihedral angles of P are submultiples of π. There is no complete classification of hyperbolic Coxeter polytopes. In [6] Vinberg proved that there are no compact hyperbolic Coxeter polytopes in IH...

The census provides a basic collection of noncompact hyperbolic 3-manifolds of finite volume. It contains descriptions of all hyperbolic 3manifolds obtained by gluing the faces of at most seven ideal tetrahedra. Additionally, various geometric and topological invariants are calculated for these manifolds. The findings are summarized and a listing of all manifolds appears in the microfiche suppl...

Article history: Received 31 July 2015 Received in revised form 17 December 2015 Accepted 10 January 2016 Available online 12 January 2016

Certainly the case of manifolds is the most interesting and most complicated. The main aim of this paper is to show that Conjecture I is not absolutely groundless. First our results deal with reflection orbifolds. In this case we will prove Conjecture 1 and find that R(Tq(M), 4) is a smooth manifold of dimension ( f -4 ) , where f is the number of "faces" of the reflection orbifold (Theorem 1)....

We compute the space of L 2 harmonic forms (outside the middle degrees) on negatively curved Kähler manifolds of finite volume.