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，技术上称为欧几里德几何、球面几何或双曲几何。三维物体通常也有几种类型，包括欧几里得、球面和双曲，以及其他一些类型。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。这个项目是关于三维双曲空间的几何。它的广泛目标是从基本的构建块构建完整的几何模型，完成该主题的长期探索。除了其核心研究目标外，该项目还通过继续指导学生、倡导代表性不足的群体以及一些推广和服务活动的计划，为专业和科学进步服务。该项目关注双曲三维流形的几何及其变形。该项目的中心是瑟斯顿的蒙皮映射的研究，这是一个全纯函数的Teichmüller空间相关联的双曲3-流形的无限体积。这个映射的图像的大小和形状对双曲流形的底层几何有影响，该项目的目标是根据底层流形边界的拓扑来限制图像的大小，而不依赖于底层流形的拓扑。利用这一点，该项目旨在构建双曲三维流形的统一模型。该方法所涉及的技术具有更广泛的兴趣，该项目旨在使用这些技术建立关于双曲流形几何可扩展性的新定理。该项目还旨在建立新的通用双曲型Dehn填充定理。该项目的新技术也将旨在建立存在的双曲积分同源球的大injectivity radiation.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。