The Geometry of Hyperbolic 3-Manifolds

表面是几何中的基本对象，通常有三种类型：平面、圆形或马鞍形，几何更专业地称为欧几里得、球形或双曲。三维物体通常也有几种风格，包括欧几里得、球形和双曲，以及其他一些。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建立完整的几何模型，完成在该主题的一长串调查。除了其核心研究目标外，该项目还通过计划继续指导学生，为代表性不足的群体辩护，以及一些外展和服务活动，为专业和科学进步服务。该项目是有关的几何双曲3流形及其变形。该项目主要研究Thurston的蒙皮映射，这是一个与无限体积的双曲3流形相关的teichm<e:1>空间上的全纯函数。该地图图像的大小和形状对双曲流形的基础几何形状有影响，该项目旨在根据基础流形边界的拓扑来约束图像的大小，而不依赖于基础流形的拓扑。利用这一点，本课题旨在构造双曲型3流形的一致模型。该方法中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不灵活性的新定理。该项目还旨在建立新的通用双曲Dehn填充定理。该项目的新技术还将旨在确定大注入半径双曲积分同调球的存在性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.