The Geometry of Hyperbolic 3-Manifolds

双曲3流形的几何

• 批准号: 2202718
• 负责人: AUTUMN KENT
• 金额: $ 43.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202718&HistoricalAwards=false
• 关键词: Geometry; Hyperbolic; Manifolds

Surfaces are fundamental objects in geometry and typically come in three flavors: flat, round, or saddle shaped, geometries more technically referred to as Euclidean, spherical, or hyperbolic. Three-dimensional objects, too, typically come in one of a few flavors, including Euclidean, spherical, and hyperbolic, plus a few others. In both two and three dimensions, the hyperbolic geometries are the most common and currently the most studied. This project is concerned with the geometry of 3-dimensional hyperbolic spaces. Its broad aim is to build complete models of the geometry from fundamental building blocks, completing a long line of inquiry in the subject. In addition to its core research objectives, the project serves the profession and the advancement of science through plans for continued mentoring of students, advocacy for underrepresented groups, and several outreach and service activities. The project is concerned with the geometry of hyperbolic 3-manifolds and their deformations. The project is centered on a study of Thurston's skinning map, which is a holomorphic function on the Teichmüller space associated to a hyperbolic 3-manifold of infinite volume. The size and shape of the image of this map has implications for the underlying geometry of the hyperbolic manifold, and the project aims to bound the size of the image in terms of the topology of the underlying manifold's boundary, without any dependence on the topology of the underlying manifold. Using this, the project aims to construct uniform models of hyperbolic 3-manifolds. The techniques involved in the approach are of wider interest, and the project aims to use these techniques to establish new theorems on the geometric inflexibility of hyperbolic manifolds. The project also aims to establish new universal hyperbolic Dehn filling theorems. The project's new techniques will also be aimed at establishing the existence of hyperbolic integral homology spheres of large injectivity radius.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

曲面是几何学中的基本对象，通常有三种类型：平面、圆形或马鞍形，技术上称为欧几里得、球形或双曲线几何形状，三维对象通常也有以下几种类型之一：包括欧几里得几何、球面几何和双曲几何，以及其他一些几何，在二维和三维中，双曲几何是最常见的，也是目前研究最多的。其广泛目标是从基本构建模块构建完整的几何模型，完成该学科的长期探究。除了其核心研究目标外，该项目还通过持续指导的计划服务于专业和科学进步。该项目涉及双曲 3 流形的几何形状及其变形，该项目的重点是瑟斯顿蒙皮图的研究，该图是与无限体积的双曲 3 流形相关的 Teichmüller 空间该图图像的大小和形状对双曲流形的基础几何形状有影响，该项目旨在根据拓扑来限制图像的大小。该项目旨在构建双曲 3 流形的统一模型，该方法涉及的技术范围更广。该项目旨在利用这些技术建立关于双曲流形几何不灵活性的新定理该项目还旨在建立新的通用双曲德恩填充定理该项目的新技术也将旨在建立双曲积分的存在性。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。