Comptes Rendus
no. 2
Differential geometry
The hypoelliptic Laplacian, analytic torsion and Cheeger–Müller Theorem
[Laplacien hypoelliptique, torsion analytique et théorème de Cheeger–Müller]

Présenté par : Jean-Michel Bismut

Shu Shen1
1 Département de mathématique, Université Paris-Sud, bâtiment 425, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 153-156.

Lʼobjet de cette Note est de démontrer une formule reliant les métriques de Ray–Singer hypoelliptique et de Milnor sur le déterminant de la cohomologie dʼune variété riemannienne compacte par une déformation à la Witten du laplacien hypoelliptique en théorie de de Rham.

The purpose of this Note is to prove a formula relating the hypoelliptic Ray–Singer metric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.12.012
Shu Shen 1

1 Département de mathématique, Université Paris-Sud, bâtiment 425, 91405 Orsay cedex, France 
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Shu Shen. The hypoelliptic Laplacian, analytic torsion and Cheeger–Müller Theorem. Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 153-156. doi : 10.1016/j.crma.2013.12.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.012/

[1] J.-M. Bismut The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 379-476

[2] J.-M. Bismut; G. Lebeau The Hypoelliptic Laplacian and Ray–Singer Metrics, Annals of Mathematics Studies, vol. 167, Princeton University Press, Princeton, NJ, 2008

[3] J.-M. Bismut; W. Zhang An extension of a theorem by Cheeger and Müller, Astérisque, Volume 205 (1992), p. 235

[4] J.-M. Bismut; W. Zhang Milnor and Ray–Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal., Volume 4 (1994) no. 2, pp. 136-212

[5] J. Cheeger Analytic torsion and the heat equation, Ann. Math. (2), Volume 109 (1979) no. 2, pp. 259-322

[6] L. Hörmander Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967), pp. 147-171

[7] V. Mathai; D. Quillen Superconnections, Thom classes, and equivariant differential forms, Topology, Volume 25 (1986) no. 1, pp. 85-110

[8] W. Müller Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., Volume 28 (1978) no. 3, pp. 233-305

[9] D.B. Ray; I.M. Singer R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210

[10] S. Shen, Laplacien hypoelliptique, torsion analytique et théorème de Cheeger–Müller, 2013, in press.

[11] E. Witten Supersymmetry and Morse theory, J. Differ. Geom., Volume 17 (1982) no. 4, pp. 661-692

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