The Chern-Simons invariants of cone-manifolds with Whitehead link singular set

In the present article, we obtain some explicit integral formulas for the generalized Chern-Simons function I(W(α,β)) for Whitehead link cone-manifolds in the hyperbolic and spherical cases. We also give the Chern-Simons invariant for the Whitehead link orbifolds. We find a formula for the Chern-Simons invariant of n-fold coverings of the three-sphere branched over the Whitehead link.     

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group is the minimal Euler characteristic of a closed orientable 4-manifold with fundamental group . We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

     

A conformal differential invariant and the conformal rigidity of hypersurfaces

For a hypersurface of a conformal space, we introduce a conformal differential invariant , where and are the first and the second fundamental forms of connected by the apolarity condition. This invariant is called the conformal quadratic element of . The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of . The main theorem states:

Let , and let and be two nonisotropic hypersurfaces without umbilical points in a conformal space or a pseudoconformal space of signature . Suppose that there is a one-to-one correspondence between points of these hypersurfaces, and in the corresponding points of and the following condition holds: where is a mapping induced by the correspondence . Then the hypersurfaces and are conformally equivalent.

     

Equivariant Willmore surfaces in conformal homogeneous three spaces

The complete classification of homogeneous three spaces is well known for some time. Of special interest are those with rigidity four which appear as Riemannian submersions with geodesic fibres over surfaces with constant curvature. Consequently their geometries are completely encoded in two values, the constant curvature, c$c$, of the base space and the so called bundle curvature, r$r$. In this paper, we obtain the complete classification of equivariant Willmore surfaces in homogeneous three spaces with rigidity four. All these surfaces appear by lifting elastic curves of the base space. Once more, the qualitative behaviour of these surfaces is encoded in the above mentioned parameters (c,r)$(c,r)$. The case where the fibres are compact is obtained as a special case of a more general result that works, via the principle of symmetric criticality, for bundle-like conformal structures in circle bundles. However, if the fibres are not compact, a different approach is necessary. We compute the differential equation satisfied by the equivariant Willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the Euler Lagrange equation of 4r²$4r^2$-elasticae in surfaces with constant curvature, c$c$. We also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature. It allows us to give explicit parametrizations of Willmore surfaces and Willmore tori in those conformal homogeneous 3-spaces.     

On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains

This paper concerns the Chern-Simons limit for the Abelian Maxwell-Chern-Simons model on bounded domains with vanishing gauge fields. We prove that every sequence of solutions of the Maxwell-Chern-Simons equations has a subsequence converging to a solution of the Chern-Simons equation in any Ck norms. We also show that the Maxwell-Chern-Simons equations with the nontopological type boundary condition do not admit any nontrivial solutions on star-shaped domains.     

Some conformal and projective scalar invariants of Riemannian manifolds

It is proved that, on any closed oriented Riemannian n-manifold, the vector spaces of conformal Killing, Killing, and closed conformal Killing r-forms, where 1 ≤ r ≤ n ? 1, have finite dimensions t _r , k _r , and p _r , respectively. The numbers t _r are conformal scalar invariants of the manifold, and the numbers k _r and p _r are projective scalar invariants; they are dual in the sense that t _r = t _n?r and k _r = p _n?r . Moreover, an explicit expression for a conformal Killing r-form on a conformally flat Riemannian n-manifold is given.     

On a spin conformal invariant on manifolds with boundary

Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by:

The 3-manifold recognition problem

We introduce a natural Relative Simplicial Approximation Property for maps from a 2-cell to a generalized 3-manifold and prove that, modulo the Poincaré Conjecture, 3-manifolds are precisely the generalized 3-manifolds satisfying this approximation property. The central technical result establishes that every generalized 3-manifold with this Relative Simplicial Approximation Property is the cell-like image of some generalized 3-manifold having just a 0-dimensional set of nonmanifold singularities.

     

A perturbative SU(3) Casson invariant

A perturbative SU(3) Casson invariant for integral homology 3-sphere is defined. Besides being fully perturbative, it has the nice properties: (1) is an integer. (2) It is preserved under orientation change. (3) A connect sum formula. Explicit calculations of the invariant for 1/k surgery of (2, q) torus knot are presented and compared with Boden-Herald‚s different SU(3) generalization of Casson‚s invariant. For those cases computed, the invariant defined here is a quadratic polynomial in k for k > 0 and a different quadratic polynomial for k < 0. Received: October 12, 2000     

A full characterization of invariant embeddability of unimodular planar graphs

When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.     

On essential conformal groups and a conformal invariant

We show that various conformal groups, including classical conformal diffeomorphism groups are essential. The essentiality is shown to be equivalent to the non-vanishing of a conformal cohomological invariant.     

三维Lorentz空间形式的共形群

研究了三维Lorentz空间形式R1^3,S1^2,H1^3的共形群,通过计算得到R1^3,S1^3,H1^3的共形群的具体表达形式,为进一步研究三维Lorentz空间式上的共形几何奠定基础．     

On transversely flat conformal foliations with good measures

Transversely flat conformal foliations with good transverse invariant measures are Riemannian in the sense. In particular, transversely similar foliations with good measures are transversely Riemannian as transversely -foliations.

     

A smooth invariant not a smooth function of the invariant polynomials

An explicit example is given of a smooth function invariant under a linear group action that is not a smooth function of the invariant polynomials.

     

A note on the Chern-Simons invariant of hyperbolic 3-manifolds

In this note we study how the Chern-Simons invariant behaves when two hyperbolic 3-manifolds are glued together along incompressible
thrice-punctured spheres. Such an operation produces many hyperbolic 3-manifolds with different numbers of cusps sharing the same volume and the same Chern-Simons invariant. The results in this note, combined with those of Meyerhoff and Ruberman, give an algorithm for determining the unknown constant in Neumann's simplicial formula for the Chern-Simons invariant of hyperbolic 3-manifolds.

     

On embedding an outer-planar graph in a point set

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog³n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n²) time. Our algorithm is near-optimal as there is an Ω(nlogn) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where lognd2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(nlogn) and O(n²) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(nlogn) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.     

CONSTRUCTIONS OF THE INVARIANT SETS AND INVARIANT MEASURES

In this paper, we will discuss the constructiOn problems about the invariant sets and invariant measures of continues maps~ which map complexes into themselves, using simplical approximation and Markov cbeirs. In [7], the author defined a matrix by using r-normal subdivi of the w,dimensional unit cube, considered it a Markov matrix, and constructed the invariantset and invafiant measure, In fact, the matrix he defined is not Markov matrix generally. So wewill give [7] and amendment in the last pert of this paper. We also construct an invariant set thatis the chain-recurrent set of the map by means of a non-negative matrix which only depends on themap. At hst, we will prove the higher dimension?Banach variation formuls that can simplify thetransition matrix.     

On the upper embedding of symmetric configurations with block size 3

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.     

Abelian Chern-Simons vortices and holomorphic Burgers hierarchy

We consider the Abelian Chern-Simons gauge field theory in 2+1 dimensions and its relation to the holomorphic Burgers hierarchy. We show that the relation between the complex potential and the complex gauge field as in incompressible and irrotational hydrodynamics has the meaning of the analytic Cole-Hopf transformation, linearizing the Burgers hierarchy and transforming it into the holomorphic Schrödinger hierarchy. The motion of planar vortices in Chern-Simons theory, which appear as pole singularities of the gauge field, then corresponds to the motion of zeros of the hierarchy. We use boost transformations of the complex Galilei group of the hierarchy to construct a rich set of exact solutions describing the integrable dynamics of planar vortices and vortex lattices in terms of generalized Kampe de Feriet and Hermite polynomials. We apply the results to the holomorphic reduction of the Ishimori model and the corresponding hierarchy, describing the dynamics of magnetic vortices and the corresponding lattices in terms of complexified Calogero-Moser models. We find corrections (in terms of Airy functions) to the two-vortex dynamics from the Moyal space-time noncommutativity.