[Matrice de diffusion et torsion analytique]
On considère une variété compacte ayant une partie isométrique à un cylindre fini. En faisant tendre la longueur du cylindre vers l'infini, on obtient une formule asymptotique pour le déterminant du laplacien de Hodge et un développement asymptotique de la torsion associée à la suite exacte de Mayer–Vietoris. On obtient une preuve analytique de la formule de recollement pour la torsion analytique.
We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the torsion of the corresponding Mayer–Vietoris exact sequence. As an application, we give a purely analytic proof of the gluing formula for analytic torsion.
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@article{CRMATH_2017__355_10_1089_0,
author = {Puchol, Martin and Zhang, Yeping and Zhu, Jialin},
title = {Scattering matrix and analytic torsion},
journal = {Comptes Rendus. Math\'ematique},
pages = {1089--1093},
publisher = {Elsevier},
volume = {355},
number = {10},
year = {2017},
doi = {10.1016/j.crma.2017.09.018},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.09.018/}
}
TY - JOUR AU - Puchol, Martin AU - Zhang, Yeping AU - Zhu, Jialin TI - Scattering matrix and analytic torsion JO - Comptes Rendus. Mathématique PY - 2017 SP - 1089 EP - 1093 VL - 355 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.09.018/ DO - 10.1016/j.crma.2017.09.018 LA - en ID - CRMATH_2017__355_10_1089_0 ER -
%0 Journal Article %A Puchol, Martin %A Zhang, Yeping %A Zhu, Jialin %T Scattering matrix and analytic torsion %J Comptes Rendus. Mathématique %D 2017 %P 1089-1093 %V 355 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.09.018/ %R 10.1016/j.crma.2017.09.018 %G en %F CRMATH_2017__355_10_1089_0
Puchol, Martin; Zhang, Yeping; Zhu, Jialin. Scattering matrix and analytic torsion. Comptes Rendus. Mathématique, Tome 355 (2017) no. 10, pp. 1089-1093. doi : 10.1016/j.crma.2017.09.018. http://www.numdam.org/articles/10.1016/j.crma.2017.09.018/
[1] Analytic torsion and holomorphic determinant bundles. I. Bott–Chern forms and analytic torsion, Comment. Phys.-Math., Volume 115 (1988) no. 1, pp. 49-78
[2] Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc., Volume 8 (1995) no. 2, pp. 291-363
[3] An extension of a theorem by Cheeger and Müller, Astérisque, Volume 205 (1992), p. 235 (With an appendix by François Laudenbach)
[4] An anomaly formula for Ray–Singer metrics on manifolds with boundary, Geom. Funct. Anal., Volume 16 (2006) no. 4, pp. 767-837
[5] On the gluing formula for the analytic torsion, Math. Z., Volume 273 (2013) no. 3–4, pp. 1085-1117
[6] Analytic torsion and the heat equation, Ann. of Math. (2), Volume 109 (1979) no. 2, pp. 259-322
[7] Adiabatic limits of the η-invariants. The odd-dimensional Atiyah–Patodi–Singer problem, Comment. Phys.-Math., Volume 142 (1991) no. 1, pp. 139-168
[8] Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., Volume 28 (1978) no. 3, pp. 233-305
[9] Eta invariants and manifolds with boundary, J. Differ. Geom., Volume 40 (1994) no. 2, pp. 311-377
[10] Adiabatic decomposition of the ζ-determinant and scattering theory, Mich. Math. J., Volume 54 (2006) no. 1, pp. 207-238
[11] R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210
[12] Homotopieringe und Linsenräume, Abh. Math. Semin. Univ. Hamb., Volume 11 (1935) no. 1, pp. 102-109
[13] Gluing formula of real analytic torsion forms and adiabatic limit, Isr. J. Math., Volume 215 (2017) no. 1, pp. 181-254
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