Bernd Ammann

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± ∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

ON THE GEOMETRY OF RIEMANNIAN MANIFOLDS WITH A LIE STRUCTURE AT INFINITY

We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah�Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.

COMPLEX POWERS AND NON-COMPACT MANIFOLDS

Abstract We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called “Guillemin algebras, ” whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyls formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131–160]. A Guillemin algebra can be thought of as an algebra of “abstract pseudodifferential operators.” Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,…) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for A z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melroses work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem -Δu=f, u|∂P=g on a polyhedral domain P⊂R3 using the Babuska–Kondratiev spaces Kam(P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4,33]. In particular, we show that there is no loss of Kam—regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a “trace theorem” for the restriction to the boundary of the functions in Kam(P).

Dirac-harmonic maps from index theory

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.

The Cauchy Problems for Einstein Metrics and Parallel Spinors

The restriction of a parallel spinor on some spin manifold

An obstruction for the mean curvature of a conformal immersion ⁿ→ℝⁿ⁺¹

{\mathcal{Z}}