Geometrization of 3-dimensional orbifolds, Part I: Geometry of cone manifolds

The Moduli Space of Hyperbolic Cone Structures

Let $Σ$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, Σ)$ with all cone angle less than $2π/3$ is an $m$-dimensional open cube, parameterized naturally by the $m$ cone angles. As a corollary, we will give a proof of a special case of Thurston's geometrization theorem for orbifolds.

Regenerating hyperbolic cone 3-manifolds from dimension 2

We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regeneration of the polygon, provided that the perimeter is minimal.

The distribution of geodesic excursions into the neighborhood of a cone singularity on a hyperbolic 2-orbifold

A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly simple formula.

The maximum injectivity radius of hyperbolic orbifolds

For two-dimensional orientable hyperbolic orbifolds, we show that the radius of a maximal embedded disk is greater or equal to an explicit constant ρ_T, with equality if and only if the orbifold is a sphere with three cone points of order 2, 3 and 7.

从加权orbifold研究g-向量锥的稠密性

Denseness of $g$-vector cones from weighted orbifolds

本文研究了从加权orbifold定义的簇代数中g-向量锥的闭包，发现其在特定条件下等于R^n或某个超平面的半空间。

On the volume and the Chern-Simons invariant for the $2$-bridge knot orbifolds

We extend some part of the unpublished paper written by Mednykh and Rasskazov. Using the approach indicated in this paper we derive the Riley-Mednykh polynomial for some family of the $2$-bridge knot orbifolds. As a result we obtain explicit formulae for the volume of cone-manifolds and the Chern-Simons invariant of orbifolds of the knot with Conway's notation $C(2n,4)$.

Regenerating hyperbolic cone structures from Nil

Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the deformation space constructed from the variety of representations of O-Sigma. As a corollary of this, we find examples of spherical cone manifolds with singular set a knot that are not locally rigid. Those examples have large cone angles.

The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold

Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…

Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Weiß when the cone angles are less than π. We pr…

通过ribbon范畴计数闭2-orbifold覆盖中的缠绕结构

Counting tangles in coverings of closed 2-orbifolds via ribbon categories

本文通过ribbon范畴方法，刻画了闭2-orbifold随机覆盖的小尺度典型结构，推广了Magee和Puder的结果至不可定向情形，并回答了Puder和Zimhoni的问题。

Anyons in Geometric Models of Matter

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.

Surgeries of the Gieseking hyperbolic ideal simplex manifold

In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with singularity geodesics of rotation order $k$, but as later turned out with cone angle $2(k-1)/k$. We computed also the volume of $S(k)$, tending to zero if $k$ goes to infinity. That time we naively thought that we obtained orbifolds with the above surprising property. As the reviewer of Math. Rev., Kevin P. Scannell (MR1770996 (2001g:57030)) rightly remarked, "this …

Branchfolds and rational conifolds

We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem for branchfolds and apply it to prove that a conifold can be endowed with branchfold structure if and only if it has locally finite holonomy.

Global rigidity of 3-dimensional cone-manifolds

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize our local rigidity result to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the geometric techniques developed by Boileau, Leeb and Porti to deform the cone-manifold structure to a complete non-sing…

A Geometric transition from hyperbolic to anti de Sitter geometry

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these m…

From Smooth to Almost Complex

This article mainly aims to overview the recent efforts on developing algebraic geometry for an arbitrary compact almost complex manifold. We review the results obtained by the guiding philosophy that a statement for smooth maps between smooth manifolds in terms of René Thom's transversality should also have its counterpart in pseudoholomorphic setting without requiring genericity of the almost complex structures. These include intersection of compact almost complex submanifolds, structure of pseudoholomorphic maps, zero locus of certain harmonic forms, and eigenvalues of Laplacian. In additio…