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Geometric operators on a class of
manifolds with bounded geometry

Geometric differential operators on complete, non-compact Riemannian manifolds, as the Laplacian or the Dirac operator, are and were extensively studied due to their applications to physics, geometry, and numerical analysis. But still they are not as well understood as those on closed manifolds. The main reason is that elliptic differential operators on noncompact manifolds do not share some useful properties that are true for compact manifolds, e.g. they are only Fredholm under additional conditions and the spectrum might be non-discrete.

$$L^p$$-spectra for Dirac operators on Lie manifolds. Let $$(M,g)$$ be a Riemannian manifold whose geometry at infinity is compatible with a Lie structure at infinity. On such manifolds we want to study the spectrum of geometric operators, emphasizing on the classical Dirac operator. The objectives of this subproject are:

- Decay estimates

- Limiting geometry at infinity

- Essential $$L^p$$-spectrum for geometric differential operators

- Relation to the $$L^p$$-spectrum of limiting geometries

$$L^p$$- index theorems

Boundary value problems and index theory on Lie manifolds:

- Index theory on Lie manifolds

- Classification of boundary value problems

Spectral density for geometric operators. Let $$(M,g)$$Riemannian manifold whose geometry at infinity is compatible with a Lie structure at infinity. On such manifolds we want to study the spectral density of geometric operators, emphasizing on the classical Dirac operator. The objectives of this subproject are:

- Spectral density approximations

- Influence of perturbation by potentials

- Approximating Green functions

- Spectral invariants

## Publications

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.

Related project(s):
3Geometric operators on a class of manifolds with bounded geometry

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## Team Members

Prof. Dr. Bernd Ammann
Universität Regensburg
ammann(at)berndammann.de

Albert-Ludwigs-Universität Freiburg im Breisgau