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

D. Gabai最近的工作表明，如果对双曲型3流形中短测地线周围的实体管邻域有了适当的理解，那么3流形理论中某些根本重要的开放问题将得到解决。例如，如果可以证明在任何双曲3-流形中都可以找到足够大的实体管邻域，那么就可以得出与闭双曲3-流形同伦等价的闭（不可约）3-流形本身就是双曲的。本项目的目标是深入理解双曲3流形中的固体管，并将这种理解应用于3流形理论中的基本问题。所采用的方法包括参数化最短测地线在双曲3流形中的可能方式，然后通过严格的计算机程序分析得到的参数空间。这项工作是与D. Gabai和N. Thurston共同完成的。在18世纪晚期，亨利·庞加莱开创了3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。庞加莱的一般问题是，如果我理解了基本群，我是否就完全理解了3流形？（臭名昭著的庞加莱猜想就是这个普遍问题的一个具体例子。）在20世纪70年代末和80年代初，William Thurston通过证明大多数3-流形具有双曲结构（双曲几何是最重要的非欧几里得几何），彻底改变了3-流形的研究。这就引出了一个自然而重要的庞加莱型问题：如果紧化（不可约）3-流形与紧化双曲3-流形具有相同的基本群，那么它一定也是双曲3-流形吗？这个项目的目标是肯定地回答这个问题。该方法是将问题简化为6维空间中与双曲型3流形中最短测地线周围的实体管有关的某个区域的分析。我们的计划是将这个区域分成大约10亿个子框，并分别处理每个子框。当然，这涉及到使用（严格的！）计算机程序。＊＊＊

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. ***