Inverse Spectral Problems in Riemannian Geometry

黎曼几何中的反谱问题

• 批准号: 0072534
• 负责人: Carolyn Gordon
• 金额: $ 36.29万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072534&HistoricalAwards=false
• 关键词: Inverse; Spectral; Problems; Riemannian; Geometry

AbstractAward: DMS-0072534Principal Investigator: Carolyn S. GordonInverse spectral geometry is the study of the extent to which thegeometry of a surface or, more generally, of a Riemannianmanifold can be extracted from spectral data. The primaryspectral data associated to a compact Riemannian manifold are theeigenvalues of the Laplace-Beltrami operator. The investigatorspropose to apply recently developed methods to study the extentto which the eigenvalue spectrum determines the local geometry ofa compact Riemannian manifold. They will also ask the extent towhich additional spectral data such as the spectrum of theLaplacian acting on differential forms of various degreesdetermines the geometry of the manifold. Inverse spectralproblems will be considered on Riemannian orbifolds as well as onmanifolds; orbifolds are the most tractable singular spaces. Forthe Schrodinger operator "Laplacian plus potential", the problemof recovering the potential from spectral data will be studied inthe case of line bundles over tori. In analogy to the case ofplanar domains, the lowest eigenvalue of the Laplacian on acompact Riemannian manifold may be viewed as the fundamentaltone. The question of whether random Riemann surfaces have largefirst eigenvalue will be studied using connections betweenspectra of Riemann surfaces and spectra of graphs. Fornoncompact Riemannian manifolds, the primary spectral data arethe scattering poles; the investigators expect to exhibitcontinuous families of isopolar metrics.In spectroscopy, one attempts to recover the chemical compositionor the shape of an object from the characteristic frequencies oflight or sound emitted. In the case of a vibrating membrane suchas a drumhead, viewed mathematically as a bounded region in theplane, the spectrum of characteristic frequencies corresponds tothe mathematical notion of the Laplace spectrum. The Laplacespectrum is also defined for other geometric objects calledmanifolds which arise in mathematics and physics. Theinvestigators, along with Scott Wolpert, earlier constructed thefirst examples of differently shaped drumheads (planar regions)with the same spectrum. Planar regions can differ in theirglobal shape but locally are identical; i.e., if you look at asmall piece cut out from one of the regions, you can not tellwhich region it came from. Recently, the principal investigatordeveloped methods for constructing geometric objects with thesame Laplace spectrum but which differ in their local as well asglobal shape. These methods will be used to investigate whichlocal geometric properties of manifolds are not spectrallydetermined. Additional spectral problems will also be consideredsuch as the construction of surfaces of arbitrarily large volumebut having bounded fundamental tone.

摘要：dms -0072534首席研究员：Carolyn S. gordon逆光谱几何是研究曲面几何的程度，或者更一般地说，黎曼流形的几何可以从光谱数据中提取。与紧黎曼流形相关的主要光谱数据是拉普拉斯-贝尔特拉米算子的特征值。研究人员建议应用最近发展的方法来研究特征值谱决定紧致黎曼流形局部几何的程度。他们也会问额外的光谱数据，如作用于不同程度的微分形式的拉普拉斯光谱，在多大程度上决定了流形的几何形状。逆谱问题将考虑黎曼轨道和流形；轨道是最容易处理的奇异空间。对于薛定谔算子“拉普拉斯加势”，在环面上的线束情况下，将研究从谱数据中恢复势的问题。与平面域的情况类似，紧黎曼流形上拉普拉斯的最低特征值可以看作基调。利用黎曼曲面的谱与图的谱之间的联系，研究了随机黎曼曲面是否具有最大第一特征值的问题。对于非紧致黎曼流形，主要的光谱数据是散射极点；研究人员希望展示连续的等极指标族。在光谱学中，人们试图从发射的飞行或声音的特征频率中恢复物体的化学成分或形状。在振动膜的情况下，如鼓面，在数学上被视为平面上的一个有界区域，特征频率的频谱对应于拉普拉斯谱的数学概念。拉普拉斯谱也适用于数学和物理中出现的流形等几何对象。研究人员和斯科特·沃尔珀特（Scott Wolpert）早些时候构建了具有相同光谱的不同形状鼓面（平面区域）的第一个例子。平面区域的整体形状可能不同，但局部形状相同；也就是说，如果你看着从其中一个区域切下来的一小块，你无法判断它来自哪个区域。近年来，主要研究者开发了具有相同拉普拉斯谱但局部形状和全局形状不同的几何物体的构造方法。这些方法将用于研究流形的哪些局部几何性质不是光谱确定的。附加的光谱问题也将被考虑，如任意大体积表面的构造，但具有有限的基本色调。