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

亚当斯将继续研究双曲3-流形中的尖体积。瑟斯顿（W. Thurston）曾推测，所有的3-流形都可以分解成组件，每个组件有八种几何形状中的一种。他已经在大量的3-流形中证明了这个猜想。其中七个几何形状的3流形的完整分类已经完成。最大和最丰富的情况对应于那些流形与第八几何，即双曲几何。双曲3-流形具有定义良好的体积，这是流形拓扑结构的不变量。所有有限体积双曲3-流形的体积集合是实数的良序子集。因此，确定一些由瑟斯顿提出的小体积的双曲流形是有意义的。非紧双曲3流形具有称为尖体积的附加不变量。它们提供了一种确定小体积流形的方法。这个想法已经被研究者用来确定一个或两个尖的最小双曲流形。该项目将沿着这条路线继续研究，通过扩展当前的计算机程序以创建更多的研究示例，并利用这些示例指出证明某些流形是其类中最小的流形的方法。特别地，项目的一个目标是证明数字8结补及其姊妹流形是最小体积的非紧可定向双曲3流形，也是Thurston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.