Dirac operators with $$W^{1,\infty }$$ W 1 , ∞ -potential on collapsing sequences losing one dimension in the limit

Abstract

Abstract. We study the behavior of the spectrum of the Dirac operator together with a symmetric W 1,∞ -potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin − structure on the limit space N , then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W 1,∞ -potential. In the case of an orientable limit space N , D is the spin Dirac operator D N on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D = D N ⊕ −D N . IntroductionThe structure of collapsing sequences of manifolds with bounded sectional curvature and diameter was studied in great detail by Cheeger et al. (see [6] and references therein). One of the next questions arising was how the spectrum of differential operators behaves in the limit of a collapsing sequence.As for the Laplacian on functions, Fukaya showed that if a sequence of manifolds with uniform bounded sectional curvature and diameter converges in the measured Gromov-Hausdorff-topology, then the eigenvalues of the Laplace operator converge to the eigenvalues of the Laplacian on the limit space with respect to a limit measure [8]. This is even the case if the limit space is a smooth manifold. Lott generalized this result to the Laplacian on p-forms [17,18]. Using the Bochner-type formula for Dirac operators on G-Clifford bundles on manifolds, where G ∈ {SO(n), Spin(n)}, Lott proved similar results for Dirac eigenvalues under collapse with bounded sectional curvature and diameter [16]. His results also include the Dirac operator acting on differential forms considering the measured Gromov-Hausdorff topology.In this paper, we consider sequences (M a , g a ) a∈N of (n + 1)-dimensional spin manifolds with bounded sectional curvature and diameter such that their GromovHausdorff limit (N , h) is n-dimensional. This already implies that N is a Riemannian orbifold [11, Proposition 11.5]. By restricting to the setting of spin manifolds we are able to show that the spectrum of the Dirac operator together with a uniform bounded symmetric W 1,∞ -potential converges again to the Dirac operator with S. Roos (B): Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: saroos@mpim-bonn.mpg.dehttps://doi.org/10.1007/s00229-018-1003-6 S. Roos symmetric W 1,∞ -potential on the limit space N . In this setting, the Dirac operator is taken with respect to the standard measure dvol(h). In particular, we do not need to consider the limit measure in the measured Gromov-Hausdorff topology as in [16]. This limit measure is in general different to the standard measure dvol(h). We only study the case of collapsing sequences losing one dimension in the limit because the situation is more complicated in the general case (see Remark 5.3).We use the techniques of [1], where Dirac operators on collapsing S 1 -principal bundles were studied. One of the main differences to ...

Compactness of Dirac-Einstein spin manifolds and horizontal deformations

“…In fact, if (g, ψ) ∈ Crit c (E), then the lower bound on the Q-curvature is equivalent to our condition −∆ g R g ≥ −KR g . We want to point out that the study of compactness and convergence of manifolds with underlying spinors was investigated in [19,23], also some cases of collapsing along the limit were studied in [30,31]. In our case, the functional provides more control on the spinorial component and this allow us to keep track of its limit.…”

A lift of the Seiberg-Witten equations to Kaluza-Klein 5-manifolds

“…We continue with the setup from Section 2. General arguments in [3], [8], [28] show (see also [2], [31], [32]): Proposition 4.1. Let Y 5 → X 4 be the principal circle bundle with Euler class e and Kaluza-Klein metric.…”

“…the spinors on Y that come from spinors on X. We distinguish between the cases that X is spin or non-spin (we follow the exposition in [3]; see also [1], [2], [31]).…”