A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study of combinatorial Calabi flow on ideal circle patterns.
New method finds ideal circle patterns on spheres.
The paper extends circle pattern flows to hyperbolic and Euclidean geometry.
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 characterizati…
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
New criteria for ideal circle patterns on surfaces.
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 discr…
Paper resolves spherical curvature flow problem.
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…
The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.
Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.
Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.
Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.
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). …
Proves existence of circle patterns on surfaces with cusps.
Study circle patterns on tori, linking symplectic forms and homeomorphisms.
A ``hyperideal circle pattern'' in 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. …
Paper extends circle pattern theory to obtuse angles.
Symplectic forms match on circle pattern space.
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 ca…
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,…
Unique circle patterns on spheres found for spherical conical metrics.
Study shows universal circle isomorphic to flow space ideal boundary.
Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.
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 wi…
New maps connect universal circles to ideal sphere for hyperbolic manifolds.
Paper solves degenerated circle pattern metric problem in spherical geometry.
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 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 ci…
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…
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
The paper studies ideal flows of closed curves, classifying critical points and proving flow behavior.
CMC-1 surfaces linked via Möbius transformations between circle patterns.
Paper generalizes Andreev's theorem with obtuse angles.
New method simplifies ideal curve flow with length constraint.
New patterns on spheres and hyperbolic planes described by integrable systems.
Extends circle pattern theorem to quasi-simplicial triangulations.
Survey on discrete minimal surfaces and their properties.
We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal su…
New Y-systems for Miquel dynamics are Möbius invariant.
Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to …
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 fro…
Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to h…
The paper explores universal circles for Anosov foliations and their uniqueness.
In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. We show how to find each such polyhedron in the upper half-space model by considering lines and circles in the plane. Finally, we give matrix generators in $\mathrm{…
A simple method makes Euclidean patterns look like Escher's art.
Study angle structures on 3-manifolds, linking to representation theory.