Spectral Analysis of Sub-Riemannian Structures

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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.