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

FRG：协作研究：了解小体积双曲 3 流形

• 批准号: 0554374
• 负责人: David Gabai
• 金额: $ 39.8万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554374&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Understanding; Low

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.

该重点研究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点双曲3流形可以得到完整的低体积双曲3流形。特别是，我们的目标是找到小体积闭合流形和尖点流形，并解释 SnapPea 普查在确定小体积流形方面的成功。到 1990 年代中期，闭合可定向双曲 3 流形体积的最佳下限似乎约为可能最低体积的 1/1000。然后论文“同伦双曲 3-流形是双曲”将低体积界限提高了一百倍。随后，许多作者利用这一结果来进一步改进下界估计。现在，PI 相信他们已经开发了一种基本的新工具（MOM 技术），该工具不仅可以找到低体积闭合和尖点双曲 3 流形，而且还可以详细解释为什么复杂性猜想是正确的。我们的方法是基本双曲几何、三流形拓扑、莫尔斯理论和严格计算机分析的令人满意的组合。我们的方法的实施将涉及在数学不同核心领域具有专业知识的数学家，并且对我们方法中使用的其他领域有深入的了解。 180 年前，W. Bolyai、C. F. Gauss 和 N. Lobachevsky 创建了欧几里得几何的替代几何，引发了科学思想的革命。 这种非欧几里得几何，称为双曲几何，已被证明是数学中的一个非凡工具。 例如，W. Thurston 在 1970 年代和 1980 年代的工作表明，绝大多数 3 维空间（3 流形）拥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关于底层的 3 维流形。 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 等人提出了终结层压和驯服猜想的证明。尽管取得了这些进展，并且 G.佩雷尔曼的几何化猜想可能得到了惊人的解决，但该理论最基本的要素之一仍有待理解。特别是，分析双曲 3 流形最自然的工具是使用几何来测量其大小，即计算其体积，但体积函数的行为仍然神秘：瑟斯顿证明存在最小体积双曲3 歧管，下一个最低体积，下一个最低体积，等等，但尽管经过 25 年的努力，仍然没有最终确定具有这些低体积的 3 歧管。 该提案引入了一项令人震惊的新技术——MOM 技术，PI 计划开发该技术来找到所有这些小体积流形，并解释小体积双曲流形必须具有哪些属性。