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。该项目的主要焦点是使用变分方法来构造三角剖分的闭三维流形上的常曲率度量，其中能量泛函是体积。PI从封闭的三维流形的三角剖分开始，引入了三维单形和三角剖分上的角度结构的概念。这些推广了Casson和Rivin关于具有非空环面边界的紧致3-流形的概念。四面体上的角度结构是将一个二面角分配给四面体的每条边，使每个顶点三角形成为一个球面三角形。例如，所有的球面、欧几里得和双曲四面体都是角结构。三角剖分的闭三维流形上的角结构是每个3-单形的角结构的实现，使得每条边的二面角之和是360度。固定三角流形上所有角结构的空间是开有界凸多面体。PI表明，存在一个自然的角结构体积的概念，它推广了双曲和球面的四面体体积的概念。该项目的主要关注点是体积的局部最大值。PI已经确定了体积可以连续地扩展到所有角度结构的空间的紧闭。这尤其建立了约翰·米尔纳关于经典几何中单纯形体积的猜想。还证明了如果体积在所有角结构的空间中有一个局部极大点，则流形有一个常曲率度量，或者流形包含一个非常特殊的2-球面或实射影平面。该项目的主要重点是研究所有角度结构的空间边界上的体积最大点。物理宇宙是三维的。在数学中，三维空间被称为三维流形。3-流形的研究是几何学和拓扑学中最重要的问题之一。了解三维空间的几何形状具有重要的理论和实际意义。1978年，威廉·瑟斯顿提出了一个革命性的猜想，列出了三维空间可以具有的最佳几何结构，即他猜想了三维空间可以具有的最佳形状。他还验证了一大类3维空间的猜想。R·汉密尔顿和G·佩雷尔曼最近的工作可能解决了猜想。然而，对于三维空间，如何在算法上找到三维空间的最佳形状仍然是一个悬而未决的问题。该方案的目标是构建算法。国际和平研究所已经在这个方向上取得了进展。该提议的成功完成将不仅在数学上有应用，而且在计算机图形学和计算广义相对论中也有应用。

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.