The Geometry of Hyperbolic 3-Manifolds

曲面是几何学中的基本对象，通常有三种形式：平面、圆形或马鞍形，这些几何在技术上更多地称为欧几里得、球形或双曲线。三维物体通常也有几种风格，包括欧几里得、球面和双曲线，还有其他一些。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，完成对该学科的一系列探索。除了其核心研究目标外，该项目还通过继续指导学生的计划、宣传代表性不足的群体以及几项外联和服务活动，为该专业和科学进步服务。该项目涉及双曲3-流形的几何及其变形。该项目的中心是研究瑟斯顿的蒙皮映射，它是Teichmüler空间上的一个全纯函数，它与无限体积的双曲3-流形有关。这张地图图像的大小和形状对双曲流形的底层几何图形有影响，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3-流形的统一模型。该方法所涉及的技术引起了更广泛的兴趣，该项目的目的是使用这些技术来建立关于双曲流形的几何不变性的新定理。该项目还旨在建立新的普适双曲Dehn填充定理。该项目的新技术还将致力于确定大注入量半径的双曲积分同调球的存在。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.