Spectral Analysis of Sub-Riemannian Structures

亚黎曼结构的谱分析

• 批准号: 339362576
• 负责人: Professor Dr. Wolfram Bauer
• 金额: --
• 依托单位: Institut für Analysis
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/339362576?language=en
• 关键词: Spectral; Analysis; Sub; Riemannian; Structures

This project proposes research in the intersection of Differential Geometry and Global Analysis. We aim to study relations between geometric and analytic objects induced by sub-Riemannian structures on smooth compact and non-compact manifolds. In this framework we study the inverse spectral problem of detecting geometric information from the spectrum of intrinsically induced sub-elliptic second order differential operators (sub-Laplacians). A sub-Riemannian geometry can be seen as a limit of Riemannian geometries in the Gromov-Hausdorff sense when the family of Riemannian metrics blows up transversely to a defining distribution. In this sense sub-Riemannian geometry is interpreted as a Geometry at Infinity. The project is divided into three parts. From a purely geometrical point of view we first aim to construct new sub-Riemannian structures on special manifolds which may give interesting examples in the spectral geometry. Part II and III focus on the spectral analysis. We plan to investigate sub-elliptic operators induced by sub-Riemannian stuctures on Lie groups, symmetric spaces and their quotients together with the new examples described in Part I. Limits or the asymptotic behaviour of spectral functions form important tools in our analysis and typically lead to invariants of the manifold. Among other problems we study and construct sub-Riemannian structures on exotic spheres, new isospectral (with respect to the sub-Laplacian) but non-diffeomorphic nilmanifolds, explicit expressions of the heat kernel for Laplace operators on differential forms and their deformations under adiabatic limits. This project has close links to other topics of SPP 2026 such as path integral formulas, index theory for non-elliptic operators or spectral rigidity.

该项目提出了在差异几何和全球分析的交集中的研究。我们的目的是研究由平滑紧凑和非紧凑型歧管上的亚里曼尼亚结构引起的几何和分析对象之间的关系。在此框架中，我们研究了从本质上诱导的亚涡流二阶差异算子（亚拉皮拉斯次数）中检测几何信息的反光谱问题。当Riemannian指标的家族横向以定义的分布而爆炸时，伊曼尼亚的几何形状可以看作是romov-hausdorff中riemannian几何形状的极限。从这个意义上讲，亚帝国的几何形状被解释为无穷大的几何形状。该项目分为三个部分。从纯粹的几何学角度来看，我们首先旨在在特殊歧管上构建新的亚里曼尼亚结构，这些结构可能在光谱几何形状中提供有趣的例子。第二部分和III专注于光谱分析。我们计划调查谎言群，对称空间及其商的亚纤维下算子以及第1部分的限制或光谱函数的渐近行为中所述的新示例，在我们的分析中构成了重要工具，通常会导致频谱函数的重要工具。除其他问题外，我们研究并构建了异国情调球体，新的等光谱（相对于拉普拉斯的新等谱）但非呈弱形构型尼尔曼群（Nilmanifolds）的研究和构造，在绝热限制下，laplace操作员对不同形式的laplace操作员的热内核的显式表达。该项目与SPP 2026的其他主题有密切的联系，例如路径积分公式，非椭圆运算符的索引理论或光谱刚度。