×

Gaps in the differential forms spectrum on cyclic coverings. (English) Zbl 1166.58013

Let \(M\) be a compact manifold of dimension \(n+1\) and let \(\Sigma\) be a compact oriented hypersurface in \(M\) which does not disconnect \(M\). Consider the associated cyclic covering \(Z\rightarrow \bar M\rightarrow M\). Assume the metric on \(\bar M\) to be complete.
The authors study spectral properties of the Hodge-de Rham operator \(\Delta_p\) on \(p\) forms in this setting. Previous work by G. Carron [J. Lond. Math. Soc., II. Ser. 65, No. 3, 757–768 (2002; Zbl 1027.58023)] established:
Theorem A. If \(n\equiv0\) mod \(4\) and if \(\text{sign}(\Sigma)\neq0\), then \(\text{spec}(\Delta_0^{\bar M})=[0,\infty)\).
The authors in the present paper give a related result:
Theorem B. (a) If \(p\neq{n\over2}\) and if \(p\neq {n\over2}+1\), then there is a family of periodic Riemannian metrics \(g_\varepsilon\) on \(\bar M\) such that \(\text{Spec}(\Delta_p^{\bar M})\) has \(N_\varepsilon\) gaps where \(\lim_{\varepsilon\rightarrow0}N_\varepsilon=\infty\). (b) If \(p={n\over2}\) or \(p={n\over2}+1\), the same conclusion holds provided that \(H^{n\over2}(\Sigma;\mathbb R)=0\).
The authors present results in Theorem C of the paper concerning the limiting spectrum on \(M\) for certain families of metrics \(g_\varepsilon\). They also give an extension of Theorem B to the Dirac operator. Let \(\alpha\in Z_2\) be the spin cobordism invariant.
Theorem D. Assume \(M\) is spin. There is a family \(g_\varepsilon\) of periodic Riemannian metrics on \(\bar M\) whose Dirac operator has a large number of gaps in its spectrum if and only if \(\hat A(\Sigma)=0\) in the cases \(n=4k\) or \(\alpha(\Sigma)=0\) in the case \(n=8k+1\) or \(n=8k+2\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 1027.58023
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ammann, B., Dahl, M., Humbert, E.: Surgery and harmonic spinors. Preprint arXiv:math.DG/0606224 · Zbl 1159.53021
[2] Anné C., Colbois B.: Opérateur de Hodge–Laplace sur des variétés compactes privées d’un nombre fini de boules. J. Funct. Anal. 115, 190–211 (1993) · Zbl 0791.58018 · doi:10.1006/jfan.1993.1087
[3] Anné C., Colbois B.: Spectre du Laplacien agissant sur les p-formes différentielles et écrasement d’anses. Math. Ann. 303(3), 545–573 (1995) · Zbl 0909.58054 · doi:10.1007/BF01461004
[4] Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc. 79(1), 71–99 (1976) · Zbl 0325.58015 · doi:10.1017/S0305004100052105
[5] Borisov N.V., Müller W., Schrader R.: Relative index theorems and supersymmetric scattering theory. Commun. Math. Phys. 114(3), 475–513 (1988) · Zbl 0663.58032 · doi:10.1007/BF01242140
[6] Brüning J., Seeley R.: An index theorem for first order regular singular operators. Am. J. Math. 110(4), 659–714 (1988) · Zbl 0664.58035 · doi:10.2307/2374646
[7] Carron G.: A topological criterion for the existence of half-bound states. J. Lond. Math. Soc. (2) 65(3), 757–768 (2002) · Zbl 1027.58023 · doi:10.1112/S0024610702003125
[8] Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds. Geometry of the Laplace operator, Honolulu/Hawai 1979. In: Proc. Symp. Pure Math., vol. 36, pp. 91–146 (1980)
[9] Chou A.W.: The Dirac operator on spaces with conical singularities and positive scalar curvature. Trans. Am. Math. Soc. 289, 1–40 (1985) · Zbl 0559.58024 · doi:10.1090/S0002-9947-1985-0779050-8
[10] Dodziuk J.: Eigenvalues of the Laplacian on forms. Proc. Am. Math. Soc. 85, 437–443 (1982) · Zbl 0502.58038 · doi:10.1090/S0002-9939-1982-0656119-2
[11] Dieudonné J.: Calcul infinitésimal. Hermann, Paris (1968)
[12] Hunsicker E., Mazzeo R.: Harmonic forms on manifolds with edges. Int. Math. Res. Not. 52, 3229–3272 (2005) · Zbl 1089.58007 · doi:10.1155/IMRN.2005.3229
[13] Lesch, M.: Operators of Fuchs type, conical singularitites, and asymptotic methods. Teubner-Texte zur Mathematik 136, Stuttgart (1997) · Zbl 1156.58302
[14] McDonald, P.: The Laplacian for spaces with cone-like singularities. Thesis, MIT, Cambridge (1990)
[15] Mazzeo, R.: Resolution blowups, spectral convergence and quasi-asymptotically conical spaces. Actes Colloque EDP Evian-les-Bains (2006)
[16] Post O.: Periodic manifolds with spectral gaps. J. Differ. Equat. 187, 23–45 (2003) · Zbl 1028.58029 · doi:10.1016/S0022-0396(02)00006-2
[17] Reed M., Simon B.: Methods of modern mathematical physics IV: Analysis of operators. Academic Press, New York (1978) · Zbl 0401.47001
[18] Roe, J.: Partitioning non-compact manifolds and the dual Toeplitz problem. In: Operator Algebras and Applications, pp 187–228. Cambridge University Press, Cambridge (1989)
[19] Rowlett, J.: Spectral geometry and asymptotically conic convergence. Thesis, Stanford (2006) · Zbl 1160.58307
[20] Ruberman, D., Saveliev, N.: Dirac operators on manifolds with periodic ends. Preprint arXiv:math. GT/0702271 (2007) · Zbl 1185.58018
[21] Seeley R.: Conic degeneration of the Gauss–Bonnet operator. J. Anal. Math. 59, 205–215 (1992) · Zbl 0818.58044 · doi:10.1007/BF02790226
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.