Volume and Geometric Structures on 3-Manifolds

自几何化猜想提出以来，寻找三维流形上的常曲率度量一直是低维拓扑学的核心问题之一。该项目的主要重点是使用变分方法在三角化封闭3-流形上构建恒定曲率度量，其中能量泛函是体积。从封闭3流形的三角剖分开始，PI引入了3-简单体和三角剖分上的角结构的概念。这是Casson和Rivin对具有非空环面边界的紧致3流形概念的推广。四面体上的角结构是在四面体的每个边上分配一个二面角，使每个顶点三角形成为一个球面三角形。例如，所有球面、欧几里得和双曲四面体都是角结构。三角化封闭3-流形上的角结构是用一个角结构来实现每个3-单纯形，使每条边的二面角之和为360度。固定三角形流形上所有角结构的空间是一个开有界凸多面体。PI证明了角结构有一个自然的体积概念，它将四面体的体积概念推广到双曲和球面。项目的主要焦点是体量的局部最大值点。PI已经确定了体积可以连续扩展到所有角度结构空间的紧凑闭合。特别是，这建立了约翰·米尔诺关于经典几何中单纯形体积的猜想。如果体积在所有角度结构的空间中有一个局部极大点，则流形要么有一个常曲率度量，要么包含一个非常特殊的2球或实投影平面。项目的主要重点是研究所有角度结构空间边界处的体积最大值点。物质世界是三维的。在数学中，三维空间被称为三维流形。三维流形的研究是几何和拓扑学中最重要的问题之一。了解三维空间的几何形状在理论和实践上都具有重要意义。1978年，威廉·瑟斯顿（William Thurston）提出了一个革命性的猜想，该猜想列出了三维空间可以具有的最佳几何结构，即他推测了三维空间可以具有的最佳形状。他还在一个巨大的三维空间中验证了这个猜想。R. Hamilton和G. Perelman最近的工作可能已经解决了猜想。然而，对于三维空间，如何从算法上找到三维空间的最佳形状仍然是一个开放的问题。本提案的目标是构建该算法。PI已经在这个方向上取得了进展。该提案的成功完成将不仅应用于数学，而且应用于计算机图形学和计算广义相对论。

Finding constant curvature metrics on 3-manifolds has been one of the central problems in low-dimensional topology ever since the geometrization conjecture was proposed. The main focus of the project is to construct constant curvature metrics on triangulated closed 3-manifolds using a variational approach where the energy functional is the volume. Starting with a triangulation of a closed 3-manifold, the PI introduces the concept of angle structures on 3-simplexes and on triangulations. These are generalizations of the concepts introduced by Casson and Rivin for compact 3-manifold with non-empty tori boundary. An angle structure on a tetrahedron is an assignment of a dihedral angle to each edge of the tetrahedron so that each vertex triangle becomes a spherical triangle. For instance all spherical, Euclidean and hyperbolic tetrahedra are angle structures. An angle structure on a triangulated closed 3-manifold is a realization of each 3-simplex by an angle structure so that the sum of dihedral angles at each edge is 360-degree. The space of all angle structures on a fixed triangulated manifold is an open bounded convex polytope. The PI has shown that there is a natural notion of volume of angle structures which generalizes the notion of volume of tetrahedra in hyperbolic and spherical. The main focus of the project is on the local maximum point of the volume. The PI has established that the volume can be extended continuously to the compact closure of the space of all angle structures. This, in particular, established a conjecture of John Milnor on the volume of simplexes in classical geometry. It has also been shown that if the volume has a local maximum point in the space of all angle structures, then either the manifold has a constant curvature metric, or the manifold contains a very special 2-sphere or real projective plane. The main focus of the project is to study the maximum point of the volume at the boundary of the space of all angle structures.The physical universe is 3-dimensional. In mathematics, 3-dimensional spaces are called 3-manifolds. The study of 3-manifolds is one of the most important problems in geometry and topology. Understanding geometric shapes of the 3-dimensional spaces is of vital importance theoretically and practically. In 1978, William Thurston made a revolutionary conjecture which lists the best geometry structure a 3-dimensional space can have, i.e., he conjectured the best shape a 3-dimensional space can take. He also verified the conjecture for a vast class of 3-dimensional spaces. Recent work of R. Hamilton and G. Perelman may have solved conjecture. However, for a 3-dimensional space, how to find algorithmically the best shape of a 3-dimensional space is still open. The goal of this proposal is to construct the algorithm. The PI has already made progresses in this direction. The successful completion of the proposal will have applications not only in mathematics, but also in computer graphics and computational general relativity.