$p$-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
Additional Information
- Colette Anné
- Affiliation: Laboratoire de Mathématiques Jean Leray, CNRS-Université de Nantes, Faculté des Sciences, BP 92208, 44322 Nantes, France
- Email: colette.anne@univ-nantes.fr
- Junya Takahashi
- Affiliation: Division of Mathematics, Graduate School of Information Sciences, Tôhoku University, Aoba 6-3-09, Sendai, 980-8579, Japan
- Email: t-junya@math.is.tohoku.ac.jp
- Received by editor(s): July 21, 2009
- Received by editor(s) in revised form: March 8, 2010
- Published electronically: November 10, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 1711-1735
- MSC (2010): Primary 58J50; Secondary 35P15, 53C23, 58J32
- DOI: https://doi.org/10.1090/S0002-9947-2011-05351-1
- MathSciNet review: 2869189