The geometry and spectra of hyperbolic manifolds
The geometry and spectra of hyperbolic manifolds
复制标题
双曲流形的几何和谱
DOI：
10.1007/bf02830802
复制
发表时间：
1994
期刊：
影响因子：
--
通讯作者：
P. Hislop
中科院分区：
文献类型：
--
作者：
P. Hislop
本文讨论了一类双曲流形的谱性质与几何性质之间的关系。在回顾了双曲流形的基本原理之后，讨论了紧致情形和有限体积情形的理论方面。本文的主要工作是研究一类无限体积双曲流形，它是由双曲空间Hn的离散子群<$n，即<$n =Hn/<$n，产生的。本文介绍了联合工作与RG Froese和PA佩里。对于这些无限体积双曲流形，只有很少的特征值，所以大部分的谱信息是由拉普拉斯算子的广义特征函数携带的。这些本征函数可以从绿色函数的渐近性构造。它示出了如何渐近几何的流形确定的渐近行为的绿色的功能，因此，本征函数，接近无穷大。这些信息被用来构建一个S-矩阵的流形，这是一个伪微分算子作用于一个纤维丛的部分在边界上的流形在无穷远。该算子及其逆算子作为谱参数的函数的亚纯性质被描述。导出了S-矩阵与广义本征函数之间的函数关系。这种关系以及S-矩阵及其逆矩阵的亚纯性的一个重要结论是与离散群Γ相关的爱森斯坦级数的亚纯延拓的存在性。最后，概述了最近的进展和一些开放的问题，包括散射极点的计数函数的渐近行为的讨论。
This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic spaceHnby discrete subgroups Г, i.e. ℳ =Hn/Г. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green’s function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green’s function, and hence the eigenfunctions, near infinity. This information is used to construct anS-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between theS-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of theS-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group Г. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles.