The study proves which circle bundles can be triangulated over a 3-simplex boundary.
problem Which circle bundles can be triangulated over a 3-simplex boundary?
method Proof of which circle bundles can be triangulated over a 3-simplex boundary.
result Only trivial and Hopf circle bundles can be triangulated over a 3-simplex boundary.
This paper derives formulas for Chern classes of triangulated circle bundles using combinatorial necklaces.
problem Calculating Chern classes for triangulated circle bundles over polyhedra.
method Using triangulations and necklace combinatorics, the paper derives rational parity formulas for Chern classes.
result Rational parity formulas for Chern classes of triangulated circle bundles are derived.
Minimal triangulations of circle bundles linked to circular permutations.
problem Which circle bundles can be triangulated over a given base triangulation?
method Minimal triangulations encoded by local systems of circular permutations of vertices.
result Classical Huntington transitivity axiom for cyclic orders expressed as a binary Chern cocycle.
Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon and Thurston. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularitie…
We determine the minimum number of vertices needed to provide balanced triangulations of Sd−2-bundles over S1. If d is odd and the bundle is orientable, or d is even and the bundle is non-orientable, the minimum number of vertices is 3d; otherwise, it is 3d+2. Similar results apply to al…
New formula counts triangles to compute first Chern class of circle bundles.
problem Computing the first Chern class of circle bundles combinatorially.
method New formula counts triangles in cyclic words associated with bundle.
result Formula is cyclically invariant and different from N.Mnev's.
Extends circle pattern theorem to quasi-simplicial triangulations.
problem Characterize circle patterns on quasi-simplicial triangulated surfaces.
method Use finite covering technique to reduce problem to simplicial case, prove characterization by KAT inequalities.
result Curvature image is characterized by KAT inequalities.
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.
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.
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.
In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defin…
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.
The study connects triangulated surfaces to complex projective structures and circle patterns.
problem Understanding circle patterns on complex projective tori.
method Using discrete holomorphic quadratic differentials, the approach involves cross ratio systems and Delaunay angles.
result For any triangulated torus, the projection map is a covering map with at most one branch point.
Paper introduces combinatorial Ricci flows on infinite disk triangulations.
problem Discrete Ricci flows on infinite triangulations of the open disk.
method Combinatorial Ricci flows (CRFs) in Euclidean and hyperbolic geometries.
result Existence and uniqueness of CRFs, convergence results, uniformization theorem.
We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …
This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program …
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.
For integers d≥2 and ε=0 or 1, let S1,d−1(ε) denote the sphere product S1×Sd−1 if ε=0 and the twisted Sd−1 bundle over S1 if ε=1. The main results of this paper are: (a) if d≡ε (mod 2) then S1,d−1(ε) has a unique minimal triangulation using 2d+3 …
To each once-punctured-torus bundle, Tφ, over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ(φ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the…
Closed geodesics densely cover a circle in dilation surfaces.
problem Density of closed geodesics in dilation surfaces.
method Study of Teichmüller flow and Delaunay triangulation.
result Directions of closed geodesics are dense in the circle.
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.
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…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
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 of q-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.
problem Geometry of q-rationals and their properties. method Construction and analysis of deformed Farey triangulation and deformed modular surface; definition and study of Springborn operations.
result Derivation of a formula for the q-deformed midpoint and new q-deformation of Markov numbers. The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
Minimal ideal triangulations studied for hyperbolic 3-manifolds.
problem Finding minimal triangulations of hyperbolic 3-manifolds.
method Characterization of low degree edges, layered solid torus subcomplexes, and 1-dimensional cohomology.
result Monodromy ideal triangulations of once-punctured torus bundles are minimal.
Short proofs show circle bundles are nilmanifolds and almost flat.
problem Understanding properties of circle bundles.
method Short proofs of two facts about circle bundles.
result Iterated circle bundles are nilmanifolds and almost flat.
The paper generates triangulations of 2-knot complements via spinning 1-knots.
problem Generating triangulations of 2-knot complements.
method Algorithm to generate triangulations of 2-knot complements obtained by spinning 1-knots.
result Triangulations of exteriors of 2-knots from all 1-knots with up to eight crossings.
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.
We give a complete characterization of all possible pairs (v,e), where v is the number of vertices and e is the number of edges, of any simplicial triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1 are unique among all t…
The paper bounds the complexity of certain 3D shapes.
problem Determining the complexity of 3D shapes that fold onto a circle.
method Using hyperbolic geometry and mapping class groups.
result The complexity equals the translation length of the folding action.
This paper connects veering triangulations to pseudo-Anosov flows on 3-manifolds.
problem Understanding the dynamics of pseudo-Anosov flows on 3-manifolds.
method Building a dictionary between veering triangulations and pseudo-Anosov flows, using canonical circular orders and link spaces.
result A bijection between veering triangulations and pseudo-Anosov flows on 3-manifolds is established.
Criterion for flat circle bundles using intrinsically harmonic forms.
problem Characterizing flat circle bundles.
method Criterion based on intrinsic harmonicity of a specific form.
result Flatness of a principal circle bundle is equivalent to intrinsic harmonicity of a certain form.
New flow for surfaces converges to circle packings.
problem Finding circle packings on surfaces.
method Introducing combinatorial p-th Calabi flow for triangulated surfaces. result Existence and convergence of the flow for all time.
Study and solve Ricci flow for circle bundles over surfaces.
problem Solving Ricci flow for circle bundles over surfaces.
method Normalized Ricci flow equation and asymptotic behavior analysis.
result Connections to model geometries observed in solutions.
Minimal surfaces created from tiny circle packings changes.
problem Creating minimal surfaces from circle packings.
method Parametrizing deformations of circle packings and relating them to discrete minimal surfaces.
result Every minimal surface of Koebe type can be extended to a general type minimal surface.
We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isoth…
This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this…
Paper introduces new combinatorial curvature for surfaces with circle packing metrics.
problem Global rigidity of combinatorial curvature on triangulated surfaces.
method Innovative circle packing metrics, combinatorial Ricci flow, and constant curvature metrics.
result Existence of constant curvature metrics is equivalent to flow convergence.
Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
Uniformizes compact Sasakian manifolds into circle bundles.
problem Deforming compact Sasakian manifolds to locally isomorphic forms.
method Criterion based on circle bundles of anti-canonical bundles over Hermitian symmetric spaces.
result Sasakian manifolds can be deformed to locally isomorphic forms.
Eta invariant computed for circle bundles over Fano manifolds.
problem Computing eta invariant for circle bundles over Fano manifolds.
method Using spin-c Dirac operator and adiabatic limit.
result Eta invariant computed for arbitrary adiabatic parameters.
We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxim…
Proves strong version of Hopf problem for hyperbolic circle bundles.
problem Strong version of Hopf problem for circle bundles over aspherical manifolds with hyperbolic fundamental groups.
method Analyzes circle bundles over aspherical manifolds with hyperbolic fundamental groups.
result Every self-map of non-zero degree is either homotopic to a homeomorphism or a non-trivial covering and the bundle is virtually trivial.
Consider a bundle of circles passing through 0 in 4-dimensional space. It is said to be rectifiable if there is a germ of diffeomorphism at 0 that takes all circles from our bundle to straight lines. We will give a classification of all rectifiable bundles of circles containing sufficiently many circles in general posi…
Maps asymptotically embed conic transforms from circle bundles.
problem Embedding conic transforms from circle bundles.
method Asymptotic embeddings using equivariant Szegő projectors.
result Maps embed conic transforms from circle bundles.
Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") o…