Related papers: Uniformisation of a once-punctured annulus

In this paper, we introduce the notion of "geodesic cover" for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic…

We give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on…

Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show…

Looking to the fundamental domains of space groups we can investigate in which space they can be realized. If this space is hyperbolic, then the corresponding space group is also hyperbolic. In addition to the usual methods for…

To any compact hyperbolic Riemann surface $X$, we associate a new type of automorphism group -- called its *commensurability automorphism group*, $ComAut(X)$. The members of $ComAut(X)$ arise from closed circuits, starting and ending at…

We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an…

For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This…

We prove that if $X = X_1 \times \dots \times X_n$ is a product of hyperbolic Riemann surfaces of finite type and $Y = \Omega/\Gamma$ is a complex manifold, where $\Omega$ is a bounded simply-connected domain in $\mathbb{C}^m$, then the…

We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss "bicycle curves" using the generalized isoperimetric inequalities.…

Given a generic totally real torus unknotted in the unit sphere of the complex plane, we prove the following alternative : either there exists a filling of the torus by holomorphic discs and the torus is rationally convex, or its rational…

Recent work of Chinburg, Reid, and Stover has shown that certain arithmetic and algebro-geometric properties of the character variety of a hyperbolic knot complement in the 3-sphere $M=S^3\setminus K$ yields topological and number theoretic…

We introduce two notions of hyperbolicity for not necessarily K\"ahler $n$-dimensional compact complex manifolds $X$. The first, called {\it balanced hyperbolicity}, generalises Gromov's K\"ahler hyperbolicity by means of Gauduchon's…

For each k > 0 we find an explicit function f_k such that the topology of S inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian surface (compact or noncompact) with K\geq -k^2, every point p on the surface, and every…

We introduce decorated piecewise hyperbolic and spherical surfaces and discuss their discrete conformal equivalence. A decoration is a choice of circle about each vertex of the surface. Our decorated surfaces are closely related to…

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of…

In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove…

For the moduli space of the punctured spheres, we find a new equality between two symplectic forms defined on it. Namely, by treating the elements of this moduli space as the singular Euclidean metrics on a sphere, we give an interpretation…

We study one of the central open questions in one-dimensional renormalization theory -- the conjectural universality of golden-mean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second…

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension…

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of…