arXiv research

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessar…

Paper extends circle pattern theory to obtuse angles.

Paper generalizes Andreev's theorem with obtuse angles.

This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be th…

Historical notes on Thurston's 3-manifold geometry.

We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyh…

Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angle…

Improved volume estimates for right-angled polyhedra in hyperbolic space.

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to π/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperboli…

We give a proof of the Singer conjecture (on the vanishing of reduced $\ell^2$-homology except in the middle dimension) for the Davis Complex $Σ$ associated to a Coxeter system $(W,S)$ whose nerve $L$ is a triangulation of $\mathbb{S}^2$. We show that it follows from a theorem of Andreev, which gives the necessary and …

In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Del…

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…

Introduces Coxeter polyhedra in various geometries.

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…

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

Unique circle patterns on spheres found for spherical conical metrics.

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…

The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.

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 prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of som…

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…

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…

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…

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N x N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard "supersymmetry" approach, we integrate out Grassmann …

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

Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function $x \mapsto \max\{0,x\}$. We try to understand the role of depth in such neural networks by showing size lowerbounds against such network …

AI generates theorems and proofs for training theorem provers.

Global inverse function theorem proved easily using Riemannian geometry.

A new comparison theorem for geometric spaces.

Paper develops formulas and theorems in Hermitian geometry.

The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.

Revises a theorem by Thurston, finding a counter-example and a weaker version.

Proofs for Moon's theorem and its generalization.

Analyzes Saito vanishing theorem using $L^{2}$ methods.

Investigates proving geometric theorems over complex and real numbers using tilings.

Extends symplectic reduction and theorem to Lie algebroids.

Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.

Proves Thurston's bounded image theorem for Haken manifolds.

Method upgrades limit theorems to mixing limit theorems for dynamical systems.

Formulates Index III lemma and Rauch III theorem with applications.

In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…

The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.

Proves two theorems on odd-dimensional manifolds with boundary.

Sharp convergence theorem for sphere submanifolds proved.

INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.

A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.

Abstracts a theorem for non-smooth maps in infinite dimensions.