𝑝-spectrum and collapsing of connected sums

Anné, Colette; Takahashi, Junya

doi:10.1090/S0002-9947-2011-05351-1

$p$-spectrum and collapsing of connected sums
HTML articles powered by AMS MathViewer

by Colette Anné and Junya Takahashi PDF

Trans. Amer. Math. Soc. 364 (2012), 1711-1735 Request permission

Abstract:

The goal of the present paper is to calculate the limit of the spectrum of the Hodge-de Rham operator under the perturbation of collapse of one part of a connected sum. It takes place in the general problem of blowing up conical singularities introduced by R. Mazzeo and J. Rowlett.

References

Colette Anné, Gilles Carron, and Olaf Post, Gaps in the differential forms spectrum on cyclic coverings, Math. Z. 262 (2009), no. 1, 57–90. MR 2491601, DOI 10.1007/s00209-008-0363-0
Colette Anné and Bruno Colbois, Opérateur de Hodge-Laplace sur des variétés compactes privées d’un nombre fini de boules, J. Funct. Anal. 115 (1993), no. 1, 190–211 (French, with English summary). MR 1228148, DOI 10.1006/jfan.1993.1087
Colette Anné and Bruno Colbois, Spectre du laplacien agissant sur les $p$-formes différentielles et écrasement d’anses, Math. Ann. 303 (1995), no. 3, 545–573 (French). MR 1355003, DOI 10.1007/BF01461004
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
Jochen Brüning and Robert Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), no. 4, 659–714. MR 955293, DOI 10.2307/2374646
Gilles Carron, Un Théorème de l’indice relatif, Pacific J. Math. 198 (2001), no. 1, 81–107 (French, with English summary). MR 1831973, DOI 10.2140/pjm.2001.198.81
Bruno Colbois and Ahmad El Soufi, Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’, Ann. Global Anal. Geom. 24 (2003), no. 4, 337–349. MR 2015867, DOI 10.1023/A:1026257431539
Bruno Colbois and Ahmad El Soufi, Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations, Proc. Amer. Math. Soc. 134 (2006), no. 3, 715–721. MR 2180889, DOI 10.1090/S0002-9939-05-08005-6
Yves Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv. 61 (1986), no. 2, 254–270 (French). MR 856089, DOI 10.1007/BF02621914
Yves Colin de Verdière, Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 599–615 (French). MR 932800
Y. Colin de Verdière, Sur une hypothèse de transversalité d’Arnol′d, Comment. Math. Helv. 63 (1988), no. 2, 184–193 (French). MR 948776, DOI 10.1007/BF02566761
Jozef Dodziuk, Eigenvalues of the Laplacian on forms, Proc. Amer. Math. Soc. 85 (1982), no. 3, 437–443. MR 656119, DOI 10.1090/S0002-9939-1982-0656119-2
Kenji Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), no. 3, 517–547. MR 874035, DOI 10.1007/BF01389241
S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
G. Gentile and V. Pagliara, Riemannian metrics with large first eigenvalue on forms of degree $p$, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3855–3858. MR 1277111, DOI 10.1090/S0002-9939-1995-1277111-2
Pierre Guerini, Prescription du spectre du laplacien de Hodge-de Rham, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 2, 270–303 (French, with English and French summaries). MR 2061782, DOI 10.1016/j.ansens.2003.04.005
Pierre Jammes, Minoration conforme du spectre du laplacien de Hodge-de Rham, Manuscripta Math. 123 (2007), no. 1, 15–23 (French, with English summary). MR 2300056, DOI 10.1007/s00229-007-0080-8
Pierre Jammes, Prescription du spectre du laplacien de Hodge-de Rham dans une classe conforme, Comment. Math. Helv. 83 (2008), no. 3, 521–537 (French, with English and French summaries). MR 2410778, DOI 10.4171/CMH/134
P. Jammes, Prescription de la multiplicité des valeurs propres du laplacien de Hodge–de Rham, Comment. Math. Helv. 86 no. 4 (2011), 967–984.
Matthias Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997. MR 1449639
John Lott, Collapsing and Dirac-type operators, Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), 2002, pp. 175–196. MR 1919899, DOI 10.1023/A:1016247531019
John Lott, Collapsing and the differential form Laplacian: the case of a smooth limit space, Duke Math. J. 114 (2002), no. 2, 267–306. MR 1920190, DOI 10.1215/S0012-7094-02-11424-0
John Lott, Remark about the spectrum of the $p$-form Laplacian under a collapse with curvature bounded below, Proc. Amer. Math. Soc. 132 (2004), no. 3, 911–918. MR 2019973, DOI 10.1090/S0002-9939-03-07121-1
Rafe R. Mazzeo and Richard B. Melrose, The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration, J. Differential Geom. 31 (1990), no. 1, 185–213. MR 1030670
R. Mazzeo, Resolution blowups, spectral convergence and quasi-asymptotically conical spaces, Actes Colloque EDP Evian-les-Bains, 2006.
J. Rowlett, Spectral geometry and asymptotically conic convergence, Ph.D. Dissertation, Stanford University, June 2006.
Julie Rowlett, Spectral geometry and asymptotically conic convergence, Comm. Anal. Geom. 16 (2008), no. 4, 735–798. MR 2471369, DOI 10.4310/CAG.2008.v16.n4.a2
Junya Takahashi, Collapsing of connected sums and the eigenvalues of the Laplacian, J. Geom. Phys. 40 (2002), no. 3-4, 201–208. MR 1866987, DOI 10.1016/S0393-0440(01)00033-X
Junya Takahashi, On the gap between the first eigenvalues of the Laplacian on functions and $p$-forms, Ann. Global Anal. Geom. 23 (2003), no. 1, 13–27. MR 1952856, DOI 10.1023/A:1021294732338