arXiv research

Asymptotics for equidistribution of circles on hyperbolic surfaces.

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in pairs…

Eta invariant computed for circle bundles over Fano manifolds.

This paper investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbo…

Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…

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

Let $\cT$ be Teichmüller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the …

Slim curves on 3-sphere help spherical CR uniformizations.

In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the ``original knot'' fibers over the circle then the wild knot $Λ$ also fibers over the circle. As a consequence, the universal covering of $\mathbb{S}^{3}-Λ$ is $\mathbb{R}^{3}$. We p…

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…

Study a flow preserving area of plane curves, ending in a circle.

The paper solves circle packings on surfaces with boundaries.

Study on eigenvalue distribution of correlated time series deforming the semi-circle law.

We construct Colding-Minicozzi limit minimal laminations in open domains in $\rth$ with the singular set of $C^1$-convergence being any properly embedded $C^{1,1}$-curve. By Meeks' $C^{1,1}$-regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination ${\cal L}$ is a locally finit…

The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.

Develops diffusion models for time-varying correlation on the circle.

In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit…

We will look at the entire cycle of the investment process relating to all aspects of, formulating an investment hypothesis, constructing a portfolio based on that, executing the trades to implement it, on-going risk management, periodically measuring the performance of the portfolio, and rebalancing the portfolio eith…

Study compact sequences of warped product circles over spheres with nonnegative scalar curvature.

We study surface groups $Γ$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $Γ$ so that its limit set $Λ_Γ$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - Λ_Γ) / Γ$ is a circle bundle over a surface. This circle bundl…

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). …

A simple method makes Euclidean patterns look like Escher's art.

Starting from an even definite lattice, we construct a principal circle bundle covered by a certain three-step nilpotent Lie group G. On the base space, which is again a nilmanifold, we then study the Dirac operator twisted by the associated complex line bundles. Noting that the whole situation fibers over the circle, …

New patterns on spheres and hyperbolic planes described by integrable systems.

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic…

We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.

Special Lagrangian submanifolds emerge from K3 surface collapse.

We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of $\mathbb{R}$-circles, is connected, and contains a Hopf link with three components; we also show that the fundamental group of its complement in $S^3$ …

New inequality on sphere generalizes circle inequality.

Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.

By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…

We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are…

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

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limi…

Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends toward a pair of projectively measured laminations that bind the surface, there is…

Study generates infinite circle packings with a specific property.

The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.

Solves Apollonius' problem using oriented circles and inversive geometry.

The paper explores universal circles for Anosov foliations and their uniqueness.

Link projections with the same circle arrangement can be transformed by specific moves.

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …

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

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 …

Classifies surfaces with great and small circles through each point.

Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.

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

Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.