Mathematical Sciences: Mobius Structures on Low-Dimensional Manifolds

[401778]本研究的目的是研究3流形上的莫比乌斯结构（共形平坦黎曼度量）和黎曼曲面上的复杂投影结构。在三维空间中，证明了柯伊伯关于表面上一类圆束上存在莫比乌斯结构的猜想是正确的。这些莫比乌斯结构的构造是基于S3中大约三圈半转产生的莫比乌斯群。期望对这些群（S3中三个圆的构型空间的几何）的更详细的研究将导致fenchell - nielsen在双曲曲面上的工作完全推广到Seifert 3-流形上的Moebius结构。还证明了给定小于2的任意数，任意闭3流形具有给定数的锥角的莫比乌斯锥结构。期望在以莫比乌斯结构表示的共形类中存在一个恒定标量曲率的奇异黎曼度规，并且当标量曲率为-1时，该度规是唯一的。黎曼度量的Hausdorff收敛将在莫比乌斯锥结构空间中引入一种拓扑结构。通过考虑这些奇异黎曼度量在锥角为2时的极限，可以得到3流形上的许多共形平面结构。在二维空间中，如果不存在明显的奇异点，则单调表示局部决定了穿孔表面上的拟有界复射影结构。因此，Teichmueller空间Tg，n支持n0的自然辛结构族。预计这些辛结构对fenchell - nielsen扭转向量是对偶的。对三维空间的研究，被称为流形，是像我们这样生活在三维宇宙（忽略时间）的生物的自然追求。近年来，Thurston等人的工作已经清楚地表明，理解具有局部非欧几里得几何（即双曲结构）的3-流形是理解一般3-流形的重要途径。我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。这就是冯洛这个项目的背景，他在方法上做了很多改变。他考虑曲面上的双曲结构，这是一个广泛的经典理论，关于一个复变量的函数。他对曲面上出现的对称双发生元群的类比的发现为新结果提供了非常丰富的来源，包括柯伊伯猜想的解决。很可能在这方面还有许多工作要做，这一观点的成果还没有用尽。＊＊＊

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. ***