Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

数学科学：黎曼几何中的逆谱问题

• 批准号: 9404298
• 负责人: Carolyn Gordon
• 金额: $ 10.5万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404298&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Spectral; Problems

9404298 Gordon The research supported by this award focuses on questions related to the geometry of manifolds. It concerned with the determination of the extent to which the spectrum of the Laplace- Beltrami operator of a compact Riemannian manifold determines the geometry of the manifold. Recent examples show, for instance, that the spectrum alone does not determine manifolds up to isometry. Further work will also be done in efforts to understand how manifolds can be isospectral yet not even locally isometric. Examples growing out of these examples provide excellent opportunities to identify specific local geometric invariants which are not spectrally determined. Work related to isospectral plane domains will also continue. Constructions of such domains are surprisingly simple; they arise as underlying spaces of certain orbifolds. Using this method, work will be done to establish whether or not it is possible to construct isospectral sets of non-isometric plane domains of any finite order. Finally, studies on Laplace versus length spectra of hyperbolic manifolds and on the Schrodinger operator on tori and Heisenberg manifolds will be carried out. Analysis of domains and manifolds through the study of classical differential operators defined on these spaces is now an established field of geometric analysis. One knows that geometric properties of the domains are intertwined with properties of the spectrum of the operators. Results of the last decade show that the dependence is not as simple as once thought. This work seeks to clarify the connections and broaden the scope of the theory's applications. ***

9404298 Gordon 该奖项支持的研究重点是与流形几何相关的问题。 它涉及确定紧致黎曼流形的拉普拉斯-贝尔特拉米算子的谱在多大程度上确定流形的几何形状。 例如，最近的例子表明，谱本身并不能确定达到等距的流形。 还将开展进一步的工作，以了解流形如何可以是等谱的，但甚至不是局部等距的。 从这些例子中衍生出来的例子提供了极好的机会来识别不是光谱确定的特定局部几何不变量。 与等谱平面域相关的工作也将继续进行。 此类域的构造出奇地简单；它们作为某些轨道折叠的底层空间而出现。 使用这种方法，将确定是否可以构造任何有限阶的非等距平面域的等谱集。 最后，将研究双曲流形的拉普拉斯与长度谱以及环面和海森堡流形上的薛定谔算子。 通过研究在这些空间上定义的经典微分算子来分析域和流形现在已成为几何分析的一个既定领域。 人们知道域的几何特性与算子谱的特性交织在一起。 过去十年的结果表明，依赖性并不像曾经想象的那么简单。 这项工作旨在澄清这些联系并扩大该理论的应用范围。 ***