The Dirac spectrum of Bieberbach manifolds

Pfäffle, Frank

doi:10.1016/s0393-0440(00)00005-x

Cited by 49 publications

(83 citation statements)

References 9 publications

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“…It is a well known fact (see [9], [14] or [18]) that if M is a compact flat spin manifold, the spin structures on M are in a one-to-one correspondence with group homomorphisms…”

Section: Preliminariesmentioning

confidence: 99%

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Eta Invariants and Class Numbers

Miatello¹,

Podestá²

2009

Pure and Applied Mathematics Quarterly

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“…It is a well known fact (see [9], [14] or [18]) that if M is a compact flat spin manifold, the spin structures on M are in a one-to-one correspondence with group homomorphisms…”

Section: Preliminariesmentioning

confidence: 99%

“…In [18], Pfäffle computes the eta invariants of all 3-dimensional compact flat manifolds. In [21], the authors consider the case of a family of Z nmanifolds of dimension n, arriving at an expression of the eta invariant in terms of solutions of certain congruences.…”

Section: Eta Invariants and Class Numbersmentioning

confidence: 99%

Eta Invariants and Class Numbers

Miatello¹,

Podestá²

2009

Pure and Applied Mathematics Quarterly

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“…This obstruction, discovered by Long and Reid [25], is the integrality of the eta invariant (for the signature operator) of the oriented Bieberbach manifold modeling the cusps. The eta invariant of Dirac operators on Bieberbach 3-manifolds was computed by Pfäffle [36] and may provide additional obstructions. See also [18] and references therein for an introduction to the eta invariant.…”

Section: Betti Numbers and Cusps Of Hyperbolic Manifoldsmentioning

confidence: 99%

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

Golenia

Moroianu

2011

Trans. Amer. Math. Soc.

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Abstract. We describe the spectrum of the k-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k and k − 1 de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case.In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V such that the Schrödinger operator has compact resolvent, although V tends to −∞ in most of the infinity. We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension four whose cusps are rational homology spheres.

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“…β = 0, the manifold M is Ricci flat hence flat, therefore it is isometric to the quotient \ R 3 where ⊂ Isom + (R 3 , can) = R 3 SO 3 is a discrete subgroup of orientation-preserving isometries acting freely on R 3 . In other words, M is one of the Bieberbach manifolds [21].…”

Section: Compact η-Einstein 3-dimensional Minimal Flows With Transvermentioning

confidence: 99%