Mathematical Sciences: Mobius Structures on Low-Dimensional Manifolds

9401778罗 本研究工作的目标是研究3-流形上的Moebius结构（共形平坦黎曼度量）和黎曼曲面上的复射影结构。 在三维空间中，证明了Kuiper关于曲面上一类圆丛上存在Moebius结构的猜想是正确的。 这些Moebius结构的构造是基于S3中围绕三个圆的半圈所产生的Moebius群。 预计更详细的研究这些群体（几何的配置空间的三个圆圈在S3）将导致一个全面推广的芬克尔-尼尔森的工作对双曲曲面的莫比乌斯结构塞弗特3-流形。 还证明了对于任意小于2-pi的数，任意闭3-流形都具有锥角为该数的Moebius锥结构. 在Moebius结构所表示的共形类中，期望存在一个常数量曲率的奇异黎曼度量，并且当数量曲率为-1时，该度量是唯一的. 黎曼度量的Hausdorff收敛将在Moebius锥结构空间中引入一个拓扑。 当锥角达到2 π时，通过考虑这些奇异黎曼度量的极限，可以得到三维流形上的许多共形平坦结构。 在二维空间中，证明了在无明显奇点的情况下，单值表示局部决定了穿孔曲面上的拟有界复射影结构。 因此，Teichmueller空间Tg，n支持一个自然族的辛结构， 0. 预计这些辛结构与Fenchel-Nielson扭转向量是对偶的。 三维空间的研究，称为流形，是像我们这样居住在三维宇宙（忽略时间）的生物的自然追求。 近年来，从瑟斯顿和其他人的工作中可以清楚地看到， 理解具有局部非欧几何的三维流形，即双曲结构，是理解三维流形的一条重要途径。 人们应该研究流形所能支持的双曲结构的多样性及其对称性。 这就是冯落为这个项目所做的设置，他在方法上做了很多改变。 他认为双曲结构的表面，其中有一个广泛的经典理论方面的职能，一个复杂的变量。 他发现了一个类似的两个发电机组的对称性，出现在案件的表面一直是一个非常肥沃的来源，新的成果，包括解决一个猜想的柯伊伯。 在这方面，似乎还有更多的工作要做，沿着这些路线，这一观点的成果绝没有被用尽。 ***

9401778 Luo The object of this research effort is to study Moebius structures (conformally flat Riemannian metrics) on 3-manifolds, and complex projective structures on Riemann surfaces. In dimension three, Kuiper's conjecture on the existence of Moebius structures on a class of circle bundles over surfaces was shown to be true. The construction of these Moebius structures is based on the Moebius groups generated by half-turns about three circles in S3. It is expected that more detailed study of these groups (the geometry of the configuration space of three circles in S3) will lead to a full generalization of Fenchel-Nielsen's work on hyperbolic surfaces to Moebius structures on Seifert 3-manifolds. It is also shown that given any number less than 2-pi, any closed 3-manifold has Moebius cone structure with cone angle the given number. It is expected that there is a singular Riemannian metric of constant scalar curvature in the conformal class represented by the Moebius structure and that the metric is unique if the scalar curvature is -1. The Hausdorff convergence of the Riemannian metrics will then introduce a topology in the space of Moebius cone structures. Many conformally flat structures on 3-manifolds will be obtained by considering the limit of these singular Riemannian metrics as the cone angle goes to 2-pi. In dimension two, it is shown that the monodromy representation locally determines the quasi-bounded complex projective structure on punctured surfaces if there are no apparent singularities. As a consequence, the Teichmueller space Tg,n supports a natural family of symplectic structures for n 0. It is expected that these symplectic structures are dual to the Fenchel-Nielson twist vectors. The study of three-dimensional spaces, called manifolds, is a natural pursuit for creatures such as ourselves who inhabit a three-dimensional universe (neglecting time). In recent years it has become clear from work of Thurston and others that understanding 3-manifolds endowed with a locally non-Euclidean geometry known as a hyperbolic structure is an important route to the understanding of 3-manifolds in general. One should look into the multiplicity of hyperbolic structures that a manifold can support and the symmetries thereof. That is the setting for this project by Feng Luo, who rings many changes on the approach. He considers hyperbolic structures on surfaces, for which there is an extensive classical theory in terms of functions of one complex variable. His discovery of an analog of the two-generator groups of symmetries that arise in the case of surfaces has been a very fertile source of new results, including the solution of a conjecture of Kuiper. It seems likely that there is still much more to be done along these lines, that the fruits of this point of view have by no means been exhausted. ***