Spectral Analysis of Sub-Riemannian Structures

这个项目提出了在微分几何和全局分析的交叉点上的研究。我们的目的是研究光滑紧流形和非紧流形上由次黎曼结构引起的几何对象和解析对象之间的关系。在这个框架下，我们研究了从本征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格罗莫夫-豪斯多夫意义下的黎曼几何的极限，当黎曼度量族横向爆炸到定义分布时。在这个意义上，次黎曼几何被解释为无穷大的几何。该项目分为三个部分。从纯几何的角度出发，我们首先在特殊流形上构造新的次黎曼结构，这可能在谱几何中给出有趣的例子。第二部分和第三部分重点介绍了光谱分析。我们计划研究由李群、对称空间及其商上的次黎曼结构诱导的次椭圆算子，以及第一部分中描述的新例子。谱函数的极限或渐近行为是我们分析的重要工具，并且通常导致流形的不变量。在其他问题中，我们研究和构造了奇异球面上的次黎曼结构、新的等谱(相对于次拉普拉斯)但非微分同胚的零流形、微分形式上拉普拉斯算子的热核的显式表达式及其绝热极限下的变形。该项目与SPP2026的其他主题有密切的联系，如路径积分公式、非椭圆算子的指数理论或谱刚性。

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.