Mathematical Sciences: Spectral Theory of Riemannian Manifolds

数学科学：黎曼流形的谱论

• 批准号: 8900219
• 负责人: Harold Donnelly
• 金额: $ 1.59万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8900219&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Theory; Riemannian

The principal investigator will study the eigenfunctions of the Laplacian operator on complete Riemannian manifolds. When the space is compact, he will provide upper and lower bounds for the Hausdorff measure of the nodal set of these eigenfunctions. In the noncompact situation, he will derive lower bounds for the decay of these eigenfunctions. Eigenfunctions of the Laplacian operator are the mathematical representations of harmonics of bells and musical instruments. Peculiar shaped bells can have an intricate assortment of harmonics; often the third or seventh harmonics dominate. In higher dimensions, the shapes of these bells are represented mathematically by Riemannian manifolds. Were these bells to flare out to infinity, the Riemannian manifold would have to be noncompact. The principal investigator will study the decay or damping out of such harmonics.

首席研究员将研究 完备黎曼流形上拉普拉斯算子。 当 空间是紧凑的，他将提供上限和下限， 这些特征函数的节点集的Hausdorff测度。 在非紧情形下，他将推导出 这些特征函数的衰减。 拉普拉斯算子的本征函数是 钟声和音乐和声的数学表示 仪器. 形状奇特的铃铛可以有一个复杂的 谐波的分类;通常是三次或七次谐波 主宰 在更高的维度，这些钟的形状是 数学上用黎曼流形表示。 是这些 当它向无穷大爆发时，黎曼流形 必须是非紧凑的。首席研究员将研究 衰减或衰减这种谐波。