Riemannian Geometry and Spectral Analysis

黎曼几何和谱分析

• 批准号: 0072154
• 负责人: John Lott
• 金额: $ 8.03万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072154&HistoricalAwards=false
• 关键词: Riemannian; Geometry; Spectral; Analysis

DMS-0072154John W. LottRiemannian geometry is the study of curved spaces. Examples of such spaces are curves and surfaces in three-dimensional flat space. Riemann showed how tomake precise the notion of curvature for a space of arbitrary dimension. Spectral analysis can be roughly characterized as the study of how a space vibrates. More precisely, to a curved space is associated a certain partial differential operator, the Laplacian. Spectral analysis is the study of how the eigenvalues of the Laplacian depend on the underlying geometry of the space.In previous work, the principal investigator obtained relationships between the spectrum of the differential form Laplacian and the geometry of the underlying space, the latter being constrained by upper bounds on its diameterand upper and lower bounds on its curvature. In particular, he characterized when there are uniform upper bounds on the j-th eigenvalue of the p-form Laplacian, and when there are small positive eigenvalues of the p-form Laplacian. He proposes to extend this work in several directions. One direction is to just assume that there is a lower bound on the curvature of the space. New issues arise in this case, as under the assumed geometric constraints, the space can ``collapse'' to a highly singular space of lower dimension. The principal investigator's previous work, in the case of upper and lower curvature bounds, also dealt with the singular spaces that arise in a collapsing limit. However, with just a lower curvature bound, the singularspaces that arise are of a different nature. He also proposes to extend the previous work in the direction of analyzing the spectrum of the Dirac operator, under the geometric assumptions of an upper bound on the diameter of the spaceand upper and lower bounds on its curvature.

DMS-0072154John W. Lott黎曼几何是弯曲空间的研究。此类空间的示例是三维平面空间中的曲线和曲面。黎曼展示了如何精确地表达任意维度空间的曲率概念。 频谱分析可以粗略地描述为对空间如何振动的研究。 更准确地说，弯曲空间与某个偏微分算子（拉普拉斯算子）相关联。 谱分析是研究拉普拉斯算子的特征值如何依赖于空间的基础几何形状。在之前的工作中，主要研究者获得了微分形式拉普拉斯算子的谱与基础空间的几何形状之间的关系，后者受到其直径上限和曲率上限和下限的约束。特别是，他表征了 p 型拉普拉斯算子的第 j 个特征值何时存在统一上限，以及何时 p 型拉普拉斯算子存在较小的正特征值。 他建议将这项工作向几个方向扩展。 一个方向是假设空间曲率存在下界。 在这种情况下出现了新的问题，因为在假设的几何约束下，空间可以“塌陷”为较低维度的高度奇异的空间。 首席研究员之前的工作，在曲率上限和下限的情况下，也处理了崩溃极限中出现的奇异空间。 然而，只有较低的曲率界限，出现的奇异空间具有不同的性质。他还建议在空间直径上限和曲率上限和下限的几何假设下，将先前的工作扩展到分析狄拉克算子谱的方向。