11 pages, 2 figures. Updated versions will be posted on http://picard.ups-tlse.fr/~schlenker Revised version: some corrections, better proof, added references
A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings.
To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequence of a more general result, describing the hyperideal circle patterns in the boundaries of geometrically finite hyperbolic 3-manifolds (for the corresponding $\C P^1$-structures). This more general statement is obtained as a consequence of a theorem of Otal \cite{otal,bonahon-otal} on the pleating laminations of the convex cores of geometrically finite hyperbolic manifolds.
We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns, on Euclidean or hyperbolic surfaces with conical singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra.
Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings. The proof uses some elementary properties of the hype…
The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the non-uniqueness coming from Möbius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the geodesic total curvature of each circle instead of the cone angles.
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variationa…
We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.
The main objective of this study is to understand how geometric hyper-ideal circle patterns can be constructed from given combinatorial angle data. We design a hybrid method consisting of a topological/deformation approach augmented with a variational principle. In this way, together with the question of characterization of hyper-ideal patterns in terms of angle data, we address their constructability via convex optimization. We presents a new proof of the main results from Jean-Marc Schlenker's work on hyper-ideal circle patterns by developing an approach that is potentially more suitable for…
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 conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper.
Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the case of obtuse exterior intersection angles.
This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns.
This paper studies circle patterns from the viewpoint of configurations. By using the topological degree theory, we extend the Koebe-Andreev-Thurston Theorem to include circle patterns with obtuse exterior intersection angles. As a consequence, we obtain a generalized Andreev's Theorem which allows obtuse dihedral angles.
In this paper we give two different proofs of Bobenko and Springborn's theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity).
This paper proves a deformation circle pattern theorem, which gives a complete description of those circle patterns with interstices in terms of the combinatorial type, the exterior intersections angles and the conformal structures of interstices. As results, the surface version of Rivin's theorem and the approximation property of packable surfaces are obtained.
Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmüller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holom…
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.
We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed b…
We prove existence and uniqueness results for patterns of circles with prescribed intersection angles in constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdi`ere's, Br"agger's and Rivin's functionals can be derived from ours.
In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for…
We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal boundary of H^2. Given two points p,q in S^1 we write pq if p and q are the endpoints of a geodesic in Gamma. (In particular pp.) We show that …
