Spectral Analysis on Riemannian Manifolds

黎曼流形的谱分析

• 批准号: 0604861
• 负责人: Zoltan Szabo
• 金额: $ 14.28万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604861&HistoricalAwards=false
• 关键词: Spectral; Analysis; Riemannian; Manifolds

This proposal is concerned with spectral analysis both on compact and open Riemannian manifolds. On compact manifolds, one of the fundamental questions is: To what extent is the geometry of Riemannian manifolds encoded in the spectra of their Laplacians? This is a very extensively investigated question in the literature. The PI is one of the initiators of spectral investigations developed on manifolds having different local geometries. Among the isospectrality examples the PI constructed the most surprising are the isospectrality families containing both homogeneous and locally inhomogeneous metrics. Although this field became very developed in the past 15 years, it is far from being a closed area. All the constructions performed so far deal with the function-spectra and nothing is known about the form-spectra. For instance, no metrics with different local geometries are known, upto this day, whichare isospectral also on forms. The extension of the isospectrality investigations to the forms and the ultimate solution of this long standing problem of finding p-isospectral compact manifolds with different local geometries belong to the main objectives of this proposal. This goal is aimed by the new explicit intertwining operator found by the PI just recently. This new technique opens up the possibility also for explicit spectrum computations, both on functions and forms. Spectral analysis on open manifols is a developing area of mathematics. Because of the infinite trace of the fundamental kernels (such as the heat-or Schroedinger-kernel) on the open manifolds, all the methods and tools applied on compact manifolds break down in the non-compact cases. The only tool by which these infinities are handled today is the so called "regularization" by which the desired finite quantities are produced by differences of infinities. This perturbative tool has been borrowed from quantum theory ("renormalization"), where the above infinities appear as infinite self mass or self charge of particles. Just recently, the PI has a new non-pertubative tool found by which these infinities can be handled on a rather wide range of Riemannian manifolds, called Zeeman manifolds. By the explicit spectrum computation, developed in isospectrality constructions, the Hilbert space of functions on a Zeeman manifold decomposes into subspaces (zones) which are invariant under the actions of the Laplacian and the natural Heisenberg group representation. Therefore, this operator can be investigated on each zone separately, meaning that important objects such as the heat flow, Schroedinger flow, partition- and zeta-function, e. t. c. can be introduced on each zone separately. In other words, a well defined zonal geometry (de Broglie geometry) can be developed, where the most surprising result is that quantities divergent on the global setting are finite on the zonal setting. Even the zonal Feynman integral is well defined. Since the Laplacian on Zeeman manifolds is nothing but the Zeeman-Hamilton operators of free charged particles, these investigations are most relevant to the quantum physics.The main focus of this proposal will be some of the fundamental questions of spectral theory on Riemannian manifolds. On compact manifolds this field might as well be called audible versus nonaudible geometry, which designation readily suggests the fundamental question of the area: To what extend is the geometry encoded into the eigenvalues of the Laplacian (spectrum) of a Riemannian manifold? A wide range of examples show that the spectrum bears just little information about the geometry. Yet, until the early 90's, a general expectation was that the spectrum does determine the local geometry. The PI was one of the first ones who disproved this conjecture and initiated spectral investigations on manifolds having different local geometries. Among the examples the PI constructed the most interesting are the isospectral metrics such that one of them is homogeneous (having a huge group of isometries acting transitively on the manifold) while the other is locally inhomogeneous (having just a "thin" group of local isometries which do not act transitively on the manifold). These examples show that one of the most important geometric data, the group of isometries, is not spectrally determined. Though this field developed very rapidly in the past 15 years, this theory has not been extended to the forms yet. This extension of the theory, including also the solution of the long standing difficult problem of finding locally non-isometric yet p-isospectral metrics, is one of the main objectives of this proposal. Spectral theory on non-compact manifolds is struggling with the infinities appearing in calculating the trace of natural kernels such as the heat kernel. Due to these infinities, all those methods break down on non-compact manifolds which are perfectly working in the compact case. This problem of infinities is parallel to the problem of infinities appearing in quantum theory. In both cases the problem is handled by a perturbative device (regularization resp. renormalization), producing the desired finite quantities by differences of infinities. As a second subject, this proposal introduces a new, natural, non-perturbative device which produces the desired finite quantities on non-compact manifolds directly. This tool is relevant also to quantum theory.

这个建议涉及紧黎曼流形和开黎曼流形上的谱分析。在紧致流形上，一个基本的问题是：黎曼流形的几何在多大程度上被编码在它们的拉普拉斯算子的谱中？这是一个在文献中被广泛研究的问题。PI是在具有不同局部几何形状的流形上开发的谱调查的发起人之一。在等谱的例子中，PI构造的最令人惊讶的是包含齐次和局部非齐次度量的等谱族。虽然这一领域在过去15年中变得非常发达，但它远不是一个封闭的领域。到目前为止，所有的构造都是处理函数谱，而对形谱一无所知。例如，没有度量与不同的本地几何是已知的，直到这一天，这也是isospectral的形式。扩展的isospectrality调查的形式和最终解决这个长期存在的问题，找到p-isospectral紧致流形与不同的局部几何属于本建议的主要目标。这个目标是由PI最近发现的新的显式交织算子实现的。这种新技术也为显式频谱计算开辟了可能性，无论是函数还是形式。开流形上的谱分析是一个正在发展的数学领域。由于基本核（例如热核或薛定谔核）在开流形上的无限迹，所有应用于紧致流形的方法和工具在非紧致情况下都失效。今天处理这些无穷大的唯一工具是所谓的“正则化”，通过这种方法，所需的有限量由无穷大的差异产生。这个微扰工具是从量子理论（“重整化”）借来的，在量子理论中，上述无穷大表现为粒子的无穷大自质量或自电荷。就在最近，PI发现了一个新的非微扰工具，通过它可以在相当广泛的黎曼流形上处理这些无穷大，称为塞曼流形。通过显式谱计算，在等谱构造中，塞曼流形上函数的希尔伯特空间分解成在拉普拉斯算子和自然海森堡群表示的作用下不变的子空间（区域）。因此，这个算子可以在每个区域上分别研究，这意味着重要的对象，如热流，薛定谔流，分区和zeta函数，e。t. C.可以分别在每个区域上引入。换句话说，一个定义明确的带状几何（布罗意几何）可以发展，其中最令人惊讶的结果是，在全球设置发散的数量是有限的带状设置。即使是带状费曼积分也有很好的定义。由于塞曼流形上的拉普拉斯算子只是自由带电粒子的塞曼-汉密尔顿算子，因此这些研究与量子物理学密切相关，主要关注黎曼流形上谱理论的一些基本问题。在紧致流形上，这个领域也可以被称为可听几何与不可听几何，这一名称很容易地暗示了这个领域的基本问题：几何在多大程度上被编码到黎曼流形的拉普拉斯（谱）的特征值中？大量的例子表明，光谱只承载了很少的几何信息。然而，直到90年代初，普遍的期望是，光谱决定局部几何。PI是第一个谁反驳了这一猜想，并开始频谱调查流形具有不同的局部几何。在PI构造的例子中，最有趣的是等谱度量，其中一个是齐次的（具有大量在流形上传递作用的等距），而另一个是局部非齐次的（只有一组“薄”的局部等距，它们不传递作用在流形上）。这些例子表明，最重要的几何数据之一，等距群，是不是光谱确定。虽然这一领域在过去的15年里发展非常迅速，但这一理论尚未扩展到形式。这种扩展的理论，包括解决长期存在的困难问题，找到当地的非等距尚未p-isospectral度量，是本建议的主要目标之一。非紧流形上的谱理论正在与计算自然核（如热核）的迹时出现的无穷大作斗争。由于这些无限性，所有这些方法都在非紧流形上失效，而这些流形在紧流形上是完美的。这个无穷大问题与量子理论中出现的无穷大问题是平行的。在这两种情况下，问题都是由微扰设备（正则化，重正化），通过无穷大的差异产生所需的有限量。作为第二个主题，这个建议介绍了一个新的，自然的，非微扰的设备，直接产生所需的有限数量的非紧流形。这个工具也与量子理论有关。