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Spectral Analysis of Sub-Riemannian Structures

We aim to study geometric aspects of sub-Riemannian structures and relations to the spectral analysis of induced differential operators (e.g. sub-Laplace operators).

The project is divided into three parts that have close relations.

Existence and construction of sub-Riemannian geometries.

We wish to study the existence of sub-Riemannian structures (also under additional conditions such as regular or trivializable) on symmetric spaces of compact and noncompact type, Lie groups and their homogeneous spaces. In particular, the notion and analysis of sub-Riemannian curvature will be of interest since it is an object which might be detected from (or is related to) the spectral data as it happens in the Riemannian setting.

Spectral analysis of geometrically defined sub-elliptic operators.

Specific questions related to the spectral analysis and geometry of (sub)-Laplace operators concern:

• (Sub)-Laplace operators on exotic spheres
• (Sub)-Laplace operator on pseudo-H-type Lie groups and compact quotients
• Path integrals and heat kernels for "higher step" nilpotent Lie groups

Heat kernel for the Laplacian on differential forms and sub-Riemannian limit

We plan to study the (sub)-Laplacian on differential forms and its heat kernel and trace in concrete cases. We plan to consider the behaviour of such objects under taking sub-Riemannian limits. Applications to $$L^2$$-invariants may be within reach.

• Heat kernel of the form Laplacian on nilpotent Lie groups
• Novikov-Shubin invariants

## Publications

Pseudo H-type Lie groups $$G_{r,s}$$ of signature (r,s) are defined via a module action of the Clifford algebra $$C\ell_{r,s}$$ on a vector space V≅$$\mathbb{R}^{2n}$$. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let $$\mathcal{N}_{r,s}$$ denote the Lie algebra corresponding to $$G_{r,s}$$. A choice of left-invariant vector fields [$$X_1, \ldots, X_{2n}$$] which generate a complement of the center of $$\mathcal{N}_{r,s}$$ gives rise to a second order operator

$$\Delta_{r,s}:=\big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ldots +X_{2n}^2 \big{)}$$

which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of $$\Delta_{r,s}$$ in the case r=0, s>0 and study their properties. In the case of r>0 we prove that $$\Delta_{r,s}$$ admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of $$\Delta_{r,s}$$ and the existence of a fundamental solution in the space of Schwartz distributions.

 Link to preprint version

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

We construct a codimension 3completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a (4n +3)-dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere S7.

 Journal Appl. Anal. 96 (2017), 2390–2407. Link to preprint version

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

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## Team Members

Prof. Dr. Wolfram Bauer