FRG: Collaborative Research: Understanding Low Volume Hyperbolic 3-Manifolds

本课题组的目标是证明以下复杂度猜想：通过填充小拓扑复杂度的尖头双曲3-流形可以得到完整的小体积双曲3-流形。特别是，我们的目标是找到小体积的闭合和尖头流形，并解释SnapPea普查在确定小体积流形方面的成功。直到20世纪90年代中期，封闭可定向双曲3-流形的最佳体积下界似乎约为可能的最低体积的1/1000。然后论文“同伦双曲3-流形是双曲的”将小体积界提高了一百倍。随后，许多作者利用这一结果进一步改进了下界估计。现在，PI相信他们已经开发出一种基本的新工具（MOM技术），它不仅可以找到小体积的闭合和尖头双曲3流形，而且还可以详细解释为什么复杂性猜想是正确的。我们的方法是一个令人满意的混合初等双曲几何，三流形拓扑，莫尔斯理论，和严格的计算机分析。我们的方法的实施将涉及在不同核心数学领域具有专业知识的数学家，以及在我们的方法中使用的其他领域的良好知识。180年前，W.波耶、C. F.高斯和N.罗巴切夫斯基开创了一场科学思想的革命，他们创造了一种替代欧几里得几何的几何学。这种非欧几里得几何，称为双曲几何，已被证明是数学中一个了不起的工具。例如，W. Thurston在20世纪70年代和80年代的工作表明，绝大多数三维空间（3流形）具有以双曲几何为模型的几何结构，并且这种几何结构可以用来回答有关潜在的三维流形的基本问题。事实上，在过去的40年里，双曲3流形一直是人们密切关注的主题，并取得了惊人的成果；最近，Y. Minsky等人对结束层压和驯服猜想的证明。尽管取得了这些进展，而且佩雷尔曼提出的几何化猜想可能得到了惊人的解决，但该理论的一个最基本的要素仍有待了解。特别是，分析双曲3-流形最自然的工具是使用几何来测量它的大小，即计算它的体积，但体积函数的行为仍然是神秘的：瑟斯顿证明了存在最小体积的双曲3-流形，以及下一个最小的体积，以及下一个最小的，等等，但尽管经过25年的努力，没有一个具有这些小体积的3-流形被最终确定。该提案引入了一项惊人的新技术——MOM技术，pi计划开发该技术来发现所有这些小体积双曲流形，并解释小体积双曲流形必须具有的特性。

The goal of this Focused Research Group is to prove the following Complexity Conjecture: that the complete low-volume hyperbolic 3-manifolds can be obtained by filling cusped hyperbolic 3-manifolds of small topological complexity. In particular, our goal is to find the low-volume closed and cusped manifolds and to explain the success of the SnapPea census in determining the low-volume manifolds. Up to the mid 1990's the best lower bounds for volume of closed orientable hyperbolic 3-manifolds appeared to be approximately 1/1000 of the likely lowest volume. Then the paper "Homotopy Hyperbolic 3-Manifolds Are Hyperbolic" improved the low-volume bounds by a factor of one hundred. Subsequently, many authors have used this result to achieve further improvements in the lower bound estimate. Now, the PI's believe they have developed a fundamental new tool (the MOM technology) which will not only find the low-volume closed and cusped hyperbolic 3-manifolds, but also explain in sharp detail why the Complexity Conjecture is correct. Our method is a satisfying mix of elementary hyperbolic geometry, 3-manifold topology, Morse Theory, and rigorous computer analysis. The implementation of our approach will involve mathematicians with expertise in different core areas of math, and with a sound knowledge of the other areas utilized in our methodology. 180 years ago, W. Bolyai, C. F. Gauss, and N. Lobachevsky started a revolution in scientific thought by creating an alternative geometry to Euclidean geometry. This non Euclidean geometry, called hyperbolic geometry, has proven to be a remarkable tool in mathematics. For example, the work of W. Thurston in the 1970's and 1980's showed that the vast majority of 3-dimensional spaces (3-manifolds) possessed geometric structures modeled on hyperbolic geometry, and that this geometric structure could be used to answer fundamental questions about the underlying 3-dimensional manifold. In fact, hyperbolic 3-manifolds have been the subject of intense scrutiny these last 40 years with striking results achieved; most recently, the proofs of the Ending Lamination and Tameness Conjectures, by Y. Minsky et al. Despite these advances and the possible spectacular resolution of the Geometrization Conjecture by G. Perelman, one of the most basic elements of the theory remains to be understood. In particular, the most natural tool for analyzing a hyperbolic 3-manifold is to use the geometry to measure its size, i.e., to compute its volume, but the behavior of the volume function remains mysterious: Thurston proved that there is a least volume hyperbolic 3-manifold, and a next lowest volume, and a next lowest, and so on, but despite 25 years of effort, none of the 3-manifolds possessing these low volumes have been conclusively identified. This proposal introduces a startling new technique--the MOM Technology--that the PIs plan to develop to find all these low-volume manifolds and to explain what properties low-volume hyperbolic manifolds must have.