Spectrum, Geometry and Topology of Complete Manifolds

完全流形的谱、几何和拓扑

• 批准号: 1105799
• 负责人: Jiaping Wang
• 金额: $ 17.34万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105799&HistoricalAwards=false
• 关键词: Spectrum; Geometry; Topology; Complete; Manifolds

The proposal concerns some aspects on the interactions among the spectrum, topology and geometry of Riemannian manifolds. One of them is the behavior of eigenfunctions, both on compact and non-compact manifolds. It aims to gain a better understanding of the structure of spectrum on complete manifolds. The study will also lead to sharp estimates of the higher eigenvalues. Another aspect is on the size of the bottom spectrum of complete manifolds. The issues to be addressed include finding an optimal upper bound in terms of the curvature lower bounds and uncovering the effect on the geometry and topology when such an upper bound is attained. Particular emphasis is on complete resolution of some of the remaining open problems from the recent studies in this direction. The principal investigator also proposes to develop a corresponding theory for the Hodge Laplacian on complete manifolds.The spectrum or the vibrating frequencies of an geometric object is closely related to its shape and structure. The proposal is to study some aspects of this relation via new mathematical formulations and approaches. Of particular interest here is to determine the size and structure of the spectrum via curvature, a quantity that can be measured locally and describes how curved the geometric object is. Successful completion of the project will provide deeper insight into this important relation. It will also have interesting applications in both physics theory and other sciences.

该建议涉及黎曼流形的谱、拓扑和几何之间的相互作用的一些方面。其中之一是紧流形和非紧流形上本征函数的行为。它的目的是为了更好地理解完备流形上的谱结构。该研究还将导致更高的特征值的尖锐估计。另一方面是关于完备流形的底谱的大小。所要解决的问题包括找到一个最佳的上界的曲率下限和揭示的效果时，达到这样的上界的几何和拓扑结构。特别强调的是，从最近的研究在这个方向上的一些剩余的开放性问题的完全解决。主要研究者还提出了完备流形上Hodge Laplacian的相应理论。几何物体的频谱或振动频率与其形状和结构密切相关。建议是通过新的数学公式和方法来研究这种关系的某些方面。这里特别感兴趣的是通过曲率来确定光谱的大小和结构，曲率是一个可以局部测量的量，它描述了几何物体的弯曲程度。该项目的成功完成将使人们更深入地了解这一重要关系。它也将在物理理论和其他科学中有有趣的应用。