Surgery and harmonic spinors

Ammann, Bernd; Dahl, Mattias; Humbert, Emmanuel

doi:10.1016/j.aim.2008.09.013

Cited by 25 publications

(101 citation statements)

References 16 publications

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“…They use also the construction of B. Ammann, M. Dahl and E. Humbert [1]. In particular, the results of Ruberman and Saveliev imply that generically, the first band of the spectrum of the Dirac operator does not touch 0; it is not a result about the presence of many gaps in the spectrum.…”

Section: 1]mentioning

confidence: 99%

“…On the other hand, if we consider a compact spin manifold M n+1 and an oriented hypersurface Σ with trivial A-invariant or trivial α-invariant, then the recent work of B. Ammann, M. Dahl and E. Humbert [1] provides a Riemannian metric h on Σ with no harmonic spinors. Then we can scale this metric so that its associated Dirac operator on Σ has no eigenvalue in a large symmetric interval.…”

Section: 1]mentioning

confidence: 99%

“…Let us sketch the ideas here. If α n (Σ) = 0 then, by the result of [1], there exists a metric h on Σ such that the corresponding Dirac operator has no harmonic spinor. We endow M with a metric such that its restriction to Σ coincides with h. Let Λ > 0 be such that the spectrum of the Dirac operator D 0 on Σ does not intersect the interval [−Λ, Λ].…”

Section: Special Case Hmentioning

confidence: 99%

“…1 We have to take care of the fact that the spaces are not all stable under the action of A. Indeed H 1 and H 2 are stable by the action of A and consequently the spectrum of A contains n 2 − p + 1 with multiplicity b p−1 (Σ) and p − n 2 with multiplicity b p (Σ) where p runs over 0, ..., n, H 5 also is stable by A, but H 3 and H 4 are not.…”

Section: The Limit Problemmentioning

confidence: 99%

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Gaps in the differential forms spectrum on cyclic coverings

2008

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Abstract. We are interested in the spectrum of the Hodge-de Rham operator on a Z-covering X over a compact manifold M of dimension n + 1. Let Σ be a hypersurface in M which does not disconnect M and such that M − Σ is a fundamental domain of the covering. If the cohomology group H n/2 (Σ) is trivial, we can construct for each N ∈ N a metric g = g N on M , such that the Hodgede Rham operator on the covering (X, g) has at least N gaps in its (essential) spectrum. If H n/2 (Σ) = 0, the same statement holds true for the Hodge-de Rham operators on p-forms provided p / ∈ {n/2, n/2 + 1}.

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Section: 1]mentioning

confidence: 99%

Section: 1]mentioning

confidence: 99%

Section: Special Case Hmentioning

confidence: 99%

Section: The Limit Problemmentioning

confidence: 99%

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Gaps in the differential forms spectrum on cyclic coverings

2008

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“…However, the converse holds true under the additional assumption that M is connected, see [4]. The proof of the converse relies on a surgery construction preserving invertibility of the Dirac operator together with Stolz's examples of manifolds with positive scalar curvature in every spin bordism class [20].…”

Section: The α-Genusmentioning

confidence: 99%

Mass endomorphism, surgery and perturbations

Ammann¹,

Dahl²,

Hermann³

et al. 2014

Ann. inst. Fourier

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Minimal kernels of Dirac operators along maps

Wittmann

2019

Mathematische Nachrichten

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Letbe a closed spin manifold and let be a closed manifold. For maps ∶ → and Riemannian metrics on and ℎ on , we consider the Dirac operator ∕ ,ℎ of the twisted Dirac bundle Σ ⊗ ℝ * TN. To this Dirac operator one can associate an index in KO −dim( ) (pt). If is 2-dimensional, one gets a lower bound for the dimension of the kernel of ∕ ,ℎ out of this index. We investigate the question whether this lower bound is obtained for generic tupels ( , , ℎ). K E Y W O R D SDirac operator, minimal kernel, spin geometry M S C ( 2 0 1 0 ) 53-C27 INTRODUCTIONLet be a closed (i.e., compact and without boundary) 2-dimensional spin manifold with a fixed spin structure and let be a closed manifold. We study the existence and genericness 1 of maps ∶ → and Riemannian metrics on and ℎ on , such that the kernel of the Dirac operator ∕ ,ℎ of the twisted Dirac bundle Σ ⊗ ℝ * TN has quaternionic dimension zero or one. Here, Σ is the usual complex spinor bundle of and ∕ ,ℎ is called the Dirac operator along the map . This problem is inherently tied to the vanishing of an index ind * TN ( ) ∈ KO −dim( ) (pt), see e.g. [18, Eq. (7.24) on p. 151], which is a generalization of Hitchin's -index [16, Section 4.2]. If is 2-dimensional, then we have ind * TN ( ) = [ dim ℍ ker ∕ ,ℎ ] ℤ 2under the isomorphism KO −2 (pt) ≅ ℤ 2 ∶= ℤ∕2ℤ, where [ ] ℤ 2 denotes the class of ∈ ℤ in ℤ 2 . Note that ind * TN ( ) is independent of the choice of the Riemannian metrics on and . It is also invariant under homotopies of . However, the index depends on the choice of spin structure on . This means in particular, thatfor any ∶ → homotopic tõand any Riemannian metric on and ℎ on . If equality holds in (1.1), then we call the kernel of ∕ ,ℎ minimal. We expect that for generic tupels ( , , ℎ) the kernel of ∕ ,ℎ is minimal.There are many results in the literature concerning the genericness of minimal kernels under the presence of an index. In [2] it is shown that for generic metrics, the dimension of the kernel of the (untwisted) Dirac operator is as small as allowed by the index theorem of Atiyah and Singer (on a closed, connected manifold). This fact generalized results in [5] and [19]. In the latter article the author also considers spin -manifolds. The dependency of the kernel of the twisted Dirac operator, where one twists

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Surgery and harmonic spinors

Cited by 25 publications

References 16 publications

Gaps in the differential forms spectrum on cyclic coverings

Gaps in the differential forms spectrum on cyclic coverings

Mass endomorphism, surgery and perturbations

Minimal kernels of Dirac operators along maps

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