Related papers: On rigidity and convergence of circle patterns

This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the…

We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is…

In this paper, we study bijections on strictly convex sets of $\mathbf R \mathbf P^n$ for $n \geq 2$ and closed convex projective surfaces equipped with the Hilbert metric that map complete geodesics to complete geodesics as sets.…

The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal structure. We give a similar uniformization for…

A contractive condition is addressed for extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. It is…

We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the $C^2$-topology, are $C^1$ conjugate.

We study the infinitesimal rigidity of equivariant minimal maps from the universal cover of a smooth oriented surface (possibly non-compact) into a Riemannian symmetric space, focusing on representations arising from cyclic harmonic…

We consider contractions of complexified real cones, as recently introduced by Rugh in [Rugh10]. Dubois [Dub09] gave optimal conditions to determine if a matrix contracts a canonical complex cone. First we generalize his results to the case…

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases…

Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains in the Riemann…

We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed…

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward…

Harmonic maps from Riemann surfaces arise from a conformally invariant variational problem. Therefore, on one hand, they are intimately connected with moduli spaces of Riemann surfaces, and on the other hand, because the conformal group is…

Almost four decades ago, Bergman and Milton independently showed that the isotropic effective electric permittivity of a two-phase composite material with a given volume fraction is constrained to lie within lens-shaped regions in the…

Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced the discrete conformal structures on surfaces in an axiomatic approach and studied its classification. In this paper, we give a full classification of the…

We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by applying round-off to planar rotations. All orbits of these maps are conjectured to be periodic. We let the angle of rotation approach pi/2, and…

The paper is devoted to an approach to the notion of the complex dilatation based on the following observations. (1) A natural measure of the distortion of the conformal structure by a real linear automorphism of the complex plane is the…

We investigate the local geometry of a pair of independent contact structures on 3-manifolds under maps that independently preserve each contact structure. We discover that such maps are homotheties on the contact 1-forms and we discover…

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a…

In this paper, we use iterations of skinning maps on Teichm\"uller spaces to study circle packings and develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning…