arXiv research

Proves rigidity of circle packings in the plane, generalizing previous work.

Paper proves circle packings converge to Riemann mapping for Jordan domains.

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…

We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.

The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.

Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the globa…

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…

The paper proves the existence of a unique circle packing on hyperbolic surfaces.

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flo…

New theorem proves rigidity of circle packings in hyperbolic geometry.

Given a triangulated surface $M$, we use Ge-Xu's $α$-flow \cite{Ge-Xu1} to deform any initial inversive distance circle packing metric to a metric with constant $α$-curvature. More precisely, we prove that the inversive distance circle packing with constant $α$-curvature is unique if $αχ(M)\leq 0$, which generalize And…

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

Proves existence of unique circle packings on polyhedral surfaces.

We study surfaces with decorations and prove uniformization in non-Euclidean geometries.

Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.

Paper proves rigidity of discrete conformal structures on polyhedral surfaces.

We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary …

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evo…

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternati…

We show that the results in \cite{Ge-Jiang1} are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a…

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

Study generates infinite circle packings with a specific property.

The paper studies circle packings using renormalization and subdivision rules.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

Paper introduces new flows to find circle packings with specific curvature.

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

Projective rigidity of circle packings on complex surfaces proved.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.

Paper constructs hyperbolic metrics using circle packings and curvature parameters.

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.

From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are giv…

Solving polynomial equations finds circle packings on surfaces.

The paper solves circle packings on surfaces with boundaries.

Circle packings on compact surfaces simplified.

The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ_g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or -1 on the surface depending on whether g=0, 1 or \ge 2 such that the surface with this metric admits a…

We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces…

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 …

Pack hyperbolic surfaces with circles or horocycles, noting symmetries.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algo…

Circle packings on translation surfaces are consistent across different surfaces.

Paper calculates ball number of links using Lorentz geometry and circle packing.

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…