Inverse Spectral Problems in Riemannian Geometry

摘要奖：DMS-0072534首席研究员：卡罗琳·S·戈登逆谱几何是研究从光谱数据中提取曲面或黎曼流形几何的程度的学科。与紧致黎曼流形相关的主谱数据是Laplace-Beltrami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的局部几何的程度。他们还将询问附加的光谱数据，如作用于不同程度的微分形式的拉普拉斯谱，在多大程度上决定了流形的几何形状。逆谱问题将被考虑在黎曼或流形上，或流形上；或或流形是最容易处理的奇异空间。对于薛定谔算符“拉普拉斯加位势”，在环面上线丛的情况下，将研究从光谱数据恢复位势的问题。与平面区域的情况类似，紧黎曼流形上拉普拉斯算子的最低本征值可视为基调。利用随机黎曼曲面的谱与图的谱之间的联系，研究随机黎曼曲面是否具有较大的第一本征值的问题。对于非紧致的黎曼流形，主要的光谱数据是散射极；研究人员期望展示连续的等极度量族。在光谱学中，人们试图从飞行或发出的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，如鼓头，在数学上被视为平面上的一个有界区域，特征频率的谱对应于拉普拉斯谱的数学概念。拉普拉斯谱也被定义为数学和物理中出现的其他几何对象，称为流形。早些时候，研究人员与斯科特·沃尔伯特一起构建了具有相同光谱的不同形状的鼓头(平面区域)的第一个例子。平面区域的整体形状可能不同，但局部是相同的；也就是说，如果你看着从其中一个区域切下的一小块，你不能说出它来自哪个区域。最近，主要的研究人员发展了构造具有相同拉普拉斯谱但其局部和全局形状不同的几何对象的方法。这些方法将被用来研究流形的哪些局部几何性质不是谱确定的。还将考虑其他谱问题，例如构造具有任意大体积但具有有界基调的曲面。

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.