03

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 show that the space introduced by Vasy in order to construct a pseudodifferential calculus adapted to the N-body problem can be obtained as the primitive ideal spectrum of one of the N-body algebras considered by Georgescu. In the process, we provide an alternative description of the iterated blow-up space of a manifold with corners with respect to a clean semilattice of adapted submanifolds (i.e. p-submanifolds). Since our constructions and proofs rely heavily on manifolds with corners and their submanifolds, we found it necessary to clarify the various notions of submanifolds of a manifold with corners.

 Journal Annales Henri Poincaré Link to preprint version Link to published version

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

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains. Our class of domains contains the class of domains with isolated oscillating conical singularities. Our results thus generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.

 Journal Comptes Rendus Mathématique Sér. I 357 487-493 (2019) Link to preprint version Link to published version

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

Let M be a smooth manifold with (smooth) boundary &part;M and bounded geometry and &part;<sub>D</sub>M &sub; &part;M be an open and closed subset.  We prove the well-posedness of the mixed Robin boundary value problem Pu = f in M, u = 0 on &part;<sub>D</sub>M, &part;<sup>P</sup><sub>&nu;</sub> u + bu = 0 on &part;M \ &part;<sub>D</sub>M under the following assumptions. First, we assume that P satisfies the strong Legendre condition (which reduces to the uniformly strong ellipticity condition in the scalar case) and that it has totally bounded coefficients (that is, that the coefficients of P and all their derivatives are bounded). Let &part;<sub>R</sub>M &sub; &part;M \ &part;<sub>D</sub>M be the set where b&ne; 0.

 Journal Rev. Roumaine Math. Pures Appl. 64 85-111 (2019) Link to preprint version Link to published version

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

Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width", that is, that the distance from any point to the boundary is bounded uniformly. Under this assumption, we prove that the Poincaré inequality for vector valued functions holds on M. We also prove a general regularity result for uniformly strongly elliptic equations and systems on general manifolds with boundary and bounded geometry. By combining the Poincaré inequality with the regularity result, we obtain-as in the classical case-that uniformly strongly elliptic equations and systems are well-posed on M in Hadamard's sense between the usual Sobolev spaces associated to the metric. We also provide variants of these results that apply to suitable mixed Dirichlet-Neumann boundary conditions. We also indicate applications to boundary value problems on singular domains.

 Journal Mathematische Nachrichten Publisher Wiley Link to preprint version Link to published version

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

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

We discuss a method to construct Dirac-harmonic maps developed by J. Jost, X. Mo and M. Zhu. The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in the publication by Jost, Mo and Zhu were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions. In several cases we show the existence of solutions.

 Journal Lett. Math. Phys. Link to preprint version Link to published version

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

• 1

Team Members

Prof. Dr. Bernd Ammann