Hyperbolicity equations for cusped 3-manifolds and volume-rigidity of repesentations
Hyperbolicity equations for cusped 3-manifolds and volume-rigidity of repesentations
复制标题
尖点 3 流形的双曲方程和表示的体积刚性
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
S. Francaviglia
中科院分区：
文献类型：
--
作者：
S. Francaviglia
研究双曲3流形最有用的工具之一是理想三角剖分技术，由瑟斯顿介绍，以了解8字形结的补数的双曲结构。如果一个3-流形具有理想三角剖分，则通过定义每个四面体上的结构，然后通过要求全局相容性，试图在流形上构造双曲结构。直双曲理想四面体参数化的复数与积极的虚部，和兼容性转化为代数方程中的parameters.In大部分的这项工作中，我们考虑广义解的兼容性方程，没有限制的虚部，我们调查这样的解决方案定义一个全球性的结构。我们开始面对，并基本上解决了在充分的一般性，类似的二维欧几里德问题。然后，我们研究明确的例子，尖3流形，表现出各种不同的现象。最后，我们引入了一定的概念的几何解决方案，我们证明了存在性和唯一性结果，这样的解决方案，我们刻画他们的体积（适当定义）holonomy。本文的最后一部分研究了双曲三维流形的特征簇上的体积函数。我们这里的主要结果是证明的刚性定理表示的最大体积。
One of the most useful tools for studying hyperbolic 3-manifolds is the technique of ideal triangulations, introduced by Thurston to understand the hyperbolic structure of the complement of the figure-eight knot. If a 3-manifold is equipped with an ideal triangulation, one tries to construct a hyperbolic structure on the manifold by defining the structure on each tetrahedron and then by requiring global compatibility. Straight hyperbolic ideal tetrahedra are parameterized by complex numbers with positive imaginary part, and compatibility translates into algebraic equations in the parameters.In most of this work we consider generalized solutions of the compatibility equations, without restrictions on the imaginary part, and we investigate which such solutions define a global structure. We begin by facing, and essentially solving in full generality, the analogous two-dimensional Euclidean problem. We then study explicit examples of cusped 3-manifold, exhibiting a variety of different phenomena. Finally, we introduce a certain notion of geometric solution, we prove existence and uniqueness results for such solutions, and we characterize them in terms of the volume of their (suitably defined) holonomy. The last part of the thesis is devoted to the study of the volume function on the character variety of a hyperbolic 3-manifold. Our main result here is the proof of a rigidity theorem for representations of maximal volume.