Paper constructs hyperbolic metrics using circle packings and curvature parameters.
problem Creating polyhedral metrics for surfaces of various topologies.
method Using circle packings and curvature parameters, the paper constructs hyperbolic polyhedral metrics.
result Unified approach to producing polyhedral metrics for surfaces of broader topological types.
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
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…
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…
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…
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
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)≤0, which generalize And…
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
The paper proves the existence of a unique circle packing on hyperbolic surfaces.
problem Proving the existence of a unique inversive distance circle packing on hyperbolic polyhedral surfaces.
method Deforming the surface by discrete Ricci flow, doing surgery by edge flipping, and using a variational principle of a convex Ricci potential.
result There exists a unique inversive distance circle packing that is discrete conformal to the original one.
Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
Study generates infinite circle packings with a specific property.
problem Generating infinite circle packings with a unique property.
method Investigates an infinite family of circle packings and uses them to create Apollonian packings.
result Created an infinite set of circle packings with the Apollonian property.
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Proves existence of unique circle packings on polyhedral surfaces.
problem Existence of unique circle packings on polyhedral surfaces with specified discrete curvature.
method Constructs diffeomorphism between fiber bundles, uses discrete Ricci flow and edge flipping.
result Proves existence of unique inversive distance circle packings.
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
Infinite circle packings on surfaces with conical singularities are possible.
problem Finding hyperbolic metrics with prescribed angles and circle packings on surfaces with punctures.
method Using infinite triangulations and hyperbolic metrics, the approach involves identifying the underlying Riemann surface and ensuring the circle packing combinatorics match the given triangulation.
result There are infinitely many conical hyperbolic structures in a conformal class with a circle packing in the combinatorics of a given triangulation.
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in (−1,1], provided an additional condition on triangle weights. Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.
problem Finding optimal radii for packing circles in various plane regions.
method Deterministic analytic formulae and recurrence relations.
result Formulated analytic formulae for 2D circle packing on various plane shapes.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Projective rigidity of circle packings on complex surfaces proved.
problem Proving rigidity of circle packings on complex projective surfaces.
method Proved projective rigidity through triangulations and complex projective structures.
result Space of circle packings is projectively rigid on complex projective surfaces.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.
problem Whether a prescribed total geodesic curvature can be realized by a degenerated circle packing.
method Introduced combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to Chow-Luo and Takatsu methods.
result Fully characterized sufficient and necessary conditions for the existence of degenerated circle packings and showed their uniqueness.
The paper studies rigid sphere packings on 3D manifolds with boundary.
problem Investigating rigid sphere packings on 3D manifolds with boundary.
method Introducing generalized sphere packings, proving rigidity, introducing combinatorial curvature flows.
result Generalized sphere packing metrics are determined by combinatorial scalar curvature.
New theorem proves convergence of various discrete conformal structures to conformal maps.
problem Proving convergence of discrete conformal structures to conformal maps.
method General theorem using piecewise linear discrete conformal mappings and Riemannian barycentric coordinates.
result Discrete conformal mappings converge to conformal maps under certain conditions.
Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.
problem Proving Doyle conjecture for hexagonal lattice circle packings.
method Using Liouville theorem of discrete harmonic functions based on logarithmic radii ratio observation.
result Proves rigidity of Doyle spirals in hexagonal lattice circle packings with bounded radii ratios.
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.
New theorem proves rigidity of circle packings in hyperbolic geometry.
problem Rigidity of circle packings in hyperbolic geometry.
method Established maximum principles and applied them to prove rigidity.
result Proved infinite rigidity of weighted Delaunay triangulations in the Poincaré disk.
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.
problem Finding circle packings on triangulated surfaces.
method Solving a system of polynomial equations associated with surface triangulations.
result Circle packings can be found by solving polynomial equations.
The paper solves circle packings on surfaces with boundaries.
problem Circle packing on surfaces with boundaries and finite genus.
method Using Thurston's algorithm and discrete Schwarz-Pick lemma.
result A unique solution to the boundary value problem exists.
Circle packings on compact surfaces simplified.
problem Simplifying circle packings on complex surfaces.
method Uniformisation of weighted maps.
result Unified approach to circle packings.
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.
problem Pack hyperbolic surfaces efficiently.
method Discuss packing with circles or horocycles, noting symmetries.
result Observations on symmetries of packed hyperbolic surfaces.
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…
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…
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…
Circle packings on translation surfaces are consistent across different surfaces.
problem Existence and consistency of circle packings on translation surfaces.
method Analysis of circle packings on translation surfaces, including strata with varying genus.
result Same circle packing can be realized on varying translation surfaces in a certain stratum.
Paper calculates ball number of links using Lorentz geometry and circle packing.
problem Calculating the minimum number of balls needed to represent a link.
method Lorentz geometry and circle packing theorem applied to ball packings.
result Shows ball(L)≤5cr(L) for any link L. 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.…
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…
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.
This paper completes the classification of discrete conformal structures on surfaces.
problem Classifying discrete conformal structures on surfaces.
method Axiomatic approach and study of existing structures.
result Find new classes of discrete conformal structures, including generalized circle packing metrics.
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.
Study circles to understand dynamics and rigidity in homogeneous spaces.
problem Understanding dynamics and rigidity in infinite-volume homogeneous spaces.
method Addressing four questions about circle packings.
result Highlighting the interplay between dynamics, geometry, and rigidity.