Mathematical Sciences: RUI: Cusp Volumes in Hyperbolic 3-Manifolds

亚当斯将继续研究双曲三维流形中的尖点体积。W·瑟斯顿猜想，所有的3-流形都可以分解成分支，每个分支都有8个几何中的一个。他已经对大类3-流形证明了这一猜想。3-流形的完整分类已经完成，其中有7个几何图形。最大和最丰富的情况对应于具有第八几何的流形，即双曲几何。双曲三维流形有一个定义良好的体积，它是流形拓扑的不变量。所有有限体积双曲3-流形的体积集是实数的良序子集。因此，确定一些低体积的双曲流形是有意义的，这是瑟斯顿提出的一个问题。非紧双曲3-流形具有称为尖点体积的附加不变量。它们提供了一种确定低流量的歧管的方法。这个想法已经被研究者用来确定一个或两个尖点的最小双曲流形。该项目将继续沿着这些路线进行研究，方法是扩展现有的计算机程序，以便创建更多用于研究的例子，并利用这些例子指出证明某些流形是其类别中最小的流形的方法。特别地，该项目的一个目的是证明8字结补及其姊妹流形是体积最小的非紧可定向双曲3-流形，这也是瑟斯顿的一个猜想。

Adams will continue the study of cusp volumes in hyperbolic 3-manifolds. It has been conjectured by W. Thurston that all 3-manifolds can be decomposed into components, each of which has one of eight geometries. He has already proven this conjecture for large classes of 3- manifolds. A complete classification of the 3-manifolds with seven of those geometries has already been completed. The largest and richest case corresponds to those manifolds with the eighth geometry, namely hyperbolic geometry. A hyperbolic 3-manifold has a well-defined volume, which is an invariant for the topology of the manifold. The set of volumes of all finite volume hyperbolic 3-manifolds is a well-ordered subset of the reals. Hence it is of interest to determine some of the hyperbolic manifolds of low volume, a problem suggested by Thurston. A noncompact hyperbolic 3-manifold has additional invariants called cusp volumes. They offer a means of determining manifolds of low volume. This idea has already been utilized by the investigator to determine the smallest hyperbolic manifolds of one or two cusps. The project will continue investigations along these lines both by extending current computer programs in order to create more examples for study and also by utilizing the examples to point the way to proofs that certain manifolds are the smallest manifolds in their classes. In particular, a goal of the project is to prove that the figure-eight knot complement and its sister manifold are the noncompact orientable hyperbolic 3-manifolds of minimal volume, also a conjecture of Thurston's.