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 prove that the computation of the Fredholm index for fully elliptic pseudodifferential operators on Lie manifolds can be reduced to the computation of the index of Dirac operators perturbed by smoothing operators. To this end we adapt to our framework ideas coming from Baum-Douglas geometric K-homology and in particular we introduce a notion of geometric cycles that can be classified into a variant of the famous geometric K-homology groups, for the specific situation here. We also define comparison maps between this geometric K-homology theory and relative K-theory.

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

We consider pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators, which are defined on the product manifold $M_1 \times M_2$ for closed manifolds $M_1$ and $M_2$. We prove a topological index theorem for Fredholm operators of tensor product type. To this end we construct a suitable double deformation groupoid and prove a Poincaré duality type result in relative $K$-theory.

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

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we derive a general index formula of the Atiyah-Singer type.

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

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):****3**Geometric operators on a class of manifolds with bounded geometry

## Team Members

**Prof. Dr. Bernd Ammann**

Project leader

Universität Regensburg

ammann(at)berndammann.de

**Dr. Karsten Bohlen**

Researcher

Universität Regensburg

karsten.bohlen(at)mathematik.uni-regensburg.de

**JProf. Dr. Nadine Große**

Project leader

Albert-Ludwigs-Universität Freiburg im Breisgau

nadine.grosse(at)math.uni-freiburg.de