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For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely…
In this paper we introduce a new methodology for smooth rigidity of Anosov diffeomorphisms based on "matching functions." The main observation is that under certain bunching assumptions on the diffeomorphism the periodic cycle functionals…
We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincar\'e-Bendixson theorem describing recurrence properties and $\omega$-limit sets of geodesics for a meromorphic connection…
In this paper some concepts of convex analysis on hyperbolic space are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex…
The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove…
We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm…
Using the gauge/gravity correspondence, we study the properties of 2-point correlation functions of finite-temperature strongly coupled gauge field theories, defined on a curved space of general spatial topology with a dual black hole…
This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly…
We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is…
We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a…
Let $\mathcal T$ be the Teichm\"{u}ller space of marked genus $g$, $n$ punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented Teichm\"{u}ller space} of marked Riemann surfaces with nodes, \cite{Abdegn, Bersdeg}.…
Geodesic currents on closed hyperbolic surfaces are measures on the unit tangent bundle invariant under geodesic flow and orientation reversal. Every geodesic current induces a dual function on curves via the geometric intersection pairing.…
The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold $M$, for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of…
Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann…
In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is…
In this paper, we explore the properties of zeta functions associated with infinite graphs of groups that arise as quotients of cuspidal tree-lattices, including all non-uniform arithmetic quotients of the tree of rank one Lie groups over…
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation,…
We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude…
We investigate models of stationary, selfgravitating, perfect-fluid tori (disks) rotating around black holes, focusing on geometric properties of spacetime. The models are constructed within the general-relativistic hydrodynamics, assuming…
We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces…