Mathematical Sciences: Solid Tubes in Hyperbolic 3-Manifolds and Applications

小行星9626561 最近的工作D。Gabai指出，某些根本上重要的开放问题，在3流形理论将得到解决，如果一个适当的理解固体管附近的短测地线在双曲3流形可以开发。 例如，如果可以证明在任何双曲三维流形中可以找到足够大的实管邻域，那么可以得出同伦等价于闭双曲三维流形的闭（不可约）三维流形本身是双曲的。 该项目的目标是深入了解双曲三维流形中的固体管，并将这种理解应用于三维流形理论中的基本问题。 所采用的方法涉及参数化的可能方式，最短的测地线可以坐在双曲3-流形，然后通过严格的计算机程序分析得到的参数空间。 本工作与D. Gabai和N.瑟斯顿 在世纪后期，Henri Poincare开创了三维流形的研究。 他最好的方法之一是分析所讨论的3-流形中的所有循环;特别是，他研究了所谓的 3-流形的基本群 庞加莱的一般性问题是， 如果我理解了基本群，我是否因此就完全理解了三维流形呢？ (The著名的庞加莱猜想就是这个一般问题的一个具体例子。） 在20世纪70年代末和80年代初，威廉·瑟斯顿（William Thurston）通过证明大多数三维流形具有双曲结构（双曲几何是最重要的非欧几何），彻底改变了三维流形的研究。 这就引出了一个自然而重要的庞加莱型问题：如果紧致（不可约）3-流形与紧致双曲3-流形具有相同的基本群，那么它也必须是双曲3-流形吗？ 这个项目的目标是肯定地回答这个问题。 方法是把问题简化为对一个 6维空间中的某个区域，该区域与双曲三维流形中最短测地线周围的实心管有关。 该计划是将这一区域分解为大约10亿个子框，并分别处理这些子框中的每一个。 当然，这涉及到使用（严格！）电脑程序 ***

9626561 Meyerhoff Recent work of D. Gabai indicates that certain fundamentally important open problems in 3-manifold theory will be solved if an appropriate understanding of solid tube neighborhoods around short geodesics in hyperbolic 3-manifolds can be developed. For example, if it can be shown that sufficiently large solid tube neighborhoods can be found in any hyperbolic 3-manifold, then it would follow that closed (irreducible) 3-manifolds that are homotopy equivalent to closed hyperbolic 3-manifolds are themselves hyperbolic. This project has as its goal to gain a deep understanding of solid tubes in hyperbolic 3-manifolds and to apply this understanding to fundamental problems in 3-manifold theory. The methods employed involve parametrizing the possible ways that shortest geodesics can sit in hyperbolic 3-manifolds and then analyzing the resulting parameter space by means of rigorous computer programs. This work is joint with D. Gabai and N. Thurston. In the late 18th century, Henri Poincare pioneered the study of 3-manifolds. One of his best methods was to analyze all loops in the 3-manifold in question; in particular, he studied the so-called fundamental group of the 3-manifold. Poincare's general question was, if I understand the fundamental group, do I thereby understand the 3-manifold completely? (The infamous Poincare Conjecture is a specific instance of this general question.) In the late 1970's and early 1980's, William Thurston revolutionized the study of 3-manifolds by showing that most 3-manifolds have hyperbolic structures (hyperbolic geometry is the most important non-Euclidean geometry). This leads to a natural and important Poincare type question: If a compact (irreducible) 3-manifold has the same fundamental group as a compact hyperbolic 3-manifold, must it be a hyperbolic 3-manifold as well? The goal of this project is to answer this question in the affirmative. The method is to reduce the question to an analysis of a certain region in a 6-dimensional space that is related to solid tubes around shortest geodesics in hyperbolic 3-manifolds. The plan is to break this region up into about a billion sub-boxes and to deal with each of these sub-boxes separately. Naturally, this involves the use of a (rigorous!) computer program. ***