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.

本项目提出了微分几何和全局分析的交叉研究。我们的目标是研究光滑紧致流形和非紧致流形上的次黎曼结构所诱导的几何对象和解析对象之间的关系。在这个框架下，我们研究了从固有诱导的次椭圆二阶微分算子（次拉普拉斯算子）的谱中检测几何信息的逆谱问题。次黎曼几何可以看作是黎曼几何在Gromov-Hausdorff意义下的极限，当黎曼度量族在定义分布的横向上爆炸时。在这个意义上，次黎曼几何被解释为无穷大几何。该项目分为三个部分。从一个纯粹的几何的角度来看，我们的第一个目标是构建新的子黎曼结构的特殊流形，这可能会给有趣的例子，在谱几何。第二部分和第三部分着重于光谱分析。我们计划研究李群、对称空间及其子空间上由次黎曼结构诱导的次椭圆算子，并结合第一部分中描述的新例子。极限或谱函数的渐近行为在我们的分析中形成重要的工具，通常会导致流形的不变量。在其他问题中，我们研究和构建子黎曼结构的奇异球，新的等谱（相对于子拉普拉斯），但非同构的nilmanifold，明确表达的热核拉普拉斯运营商的微分形式和他们的变形下绝热限制。该项目与SPP 2026的其他主题有密切联系，例如路径积分公式，非椭圆算子的指数理论或谱刚性。