no. 2
Analytic torsions on contact manifolds
[Torsions analytiques sur les variétés de contact]
Rumin, Michel1 ; Seshadri, Neil2
1 Laboratoire de Mathématiques d’Orsay CNRS et Université Paris Sud 91405 Orsay Cedex France
2 Graduate School of Mathematical Sciences The University of Tokyo 3–8– 1 Komaba, Meguro, Tokyo 150-0044 Japan
Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 727-782.

Nous définissons et étudions la torsion analytique du complexe de contact sur les variétés de contact. Nous montrons qu’elle coïncide avec la torsion de Ray–Singer sur les variétés CR de Seifert munies d’une représentation unitaire. Nous la calculons dans ces cas et l’exprimons à l’aide de propriétés dynamiques du flot de Reeb. En fait, notre fonction spectrale de torsion analytique coïncide avec une fonction zêta dynamique naturelle. Ces formules de trace «  à la Selberg  » persistent ici pour des métriques de courbure non constante sur la base.

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray–Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-like trace formulae, that hold also in variable curvature.

DOI : 10.5802/aif.2693
Classification : 58J52, 32V05, 32V20, 11M36, 37C30
Keywords: analytic torsion, contact complex, CR Seifert manifold, trace formula
Mot clés : torsion analytique, complexe de contact, variété CR de Seifert, formule de trace

Rumin, Michel 1 ; Seshadri, Neil 2

1 Laboratoire de Mathématiques d’Orsay CNRS et Université Paris Sud 91405 Orsay Cedex France
2 Graduate School of Mathematical Sciences The University of Tokyo 3–8– 1 Komaba, Meguro, Tokyo 150-0044 Japan 
@article{AIF_2012__62_2_727_0,
     author = {Rumin, Michel and Seshadri, Neil},
     title = {Analytic torsions on contact manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {727--782},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     doi = {10.5802/aif.2693},
     mrnumber = {2985515},
     zbl = {1264.58027},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2693/}
}
TY  - JOUR
AU  - Rumin, Michel
AU  - Seshadri, Neil
TI  - Analytic torsions on contact manifolds
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 727
EP  - 782
VL  - 62
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2693/
DO  - 10.5802/aif.2693
LA  - en
ID  - AIF_2012__62_2_727_0
ER  - 
%0 Journal Article
%A Rumin, Michel
%A Seshadri, Neil
%T Analytic torsions on contact manifolds
%J Annales de l'Institut Fourier
%D 2012
%P 727-782
%V 62
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2693/
%R 10.5802/aif.2693
%G en
%F AIF_2012__62_2_727_0
Rumin, Michel; Seshadri, Neil. Analytic torsions on contact manifolds. Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 727-782. doi : 10.5802/aif.2693. https://aif.centre-mersenne.org/articles/10.5802/aif.2693/

[1] Bär, C.; Moroianu, S. Heat kernel asymptotics for roots of generalized Laplacians, Internat. J. Math., Volume 14 (2003) no. 4, pp. 397-412 | DOI | MR

[2] Baston, R. J.; Eastwood, M. G. The Penrose transform: Its interaction with representation theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989 | MR | Zbl

[3] Beals, R.; Greiner, P. Calculus on Heisenberg manifolds, Annals of Mathematics Studies, 119, Princeton University Press, Princeton, NJ, 1988 | MR | Zbl

[4] Beals, R.; Greiner, P. C.; Stanton, N. K. The heat equation and geometry of CR manifolds, Bull. Amer. Math. Soc. (N.S.), Volume 10 (1984) no. 2, pp. 275-276 | DOI | MR | Zbl

[5] Belgun, F. A. Normal CR structures on S 3 , Math. Z., Volume 244 (2003) no. 1, pp. 123-151 | DOI | MR

[6] Biquard, O.; Herzlich, M. Burns-Epstein invariant for ACHE 4-manifolds, Duke Math. J., Volume 126 (2005) no. 1, pp. 53-100 | DOI | MR

[7] Biquard, O.; Herzlich, M.; Rumin, M. Diabatic limit, eta invariants and Cauchy-Riemann manifolds of dimension 3, Ann. Sci. Ecole Norm. Sup. (4), Volume 40 (2007) no. 4, pp. 589-631 | MR

[8] Bismut, J.-M. The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., Volume 18 (2005) no. 2, p. 379-476 (electronic) | DOI | MR

[9] Bismut, J.-M. Loop spaces and the hypoelliptic Laplacian, Comm. Pure Appl. Math., Volume 61 (2008) no. 4, pp. 559-593 | DOI | MR

[10] Bismut, J.-M.; Gillet, H.; C., Soulé Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys., Volume 115 (1988) no. 1, pp. 49-78 | DOI | MR | Zbl

[11] Bismut, J.-M.; Gillet, H.; C., Soulé Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Comm. Math. Phys., Volume 115 (1988) no. 2, pp. 301-351 | DOI | MR | Zbl

[12] Bismut, J.-M.; Lebeau, G. The hypoelliptic Laplacian and Ray-Singer metrics, Annals of Mathematics Studies, 167, Princeton University Press, Princeton, NJ, 2008 | MR

[13] Bismut, J.-M.; Zhang, W. An extension of a theorem by Cheeger and Müller. With an appendix by François Laudenbach, Astérisque, 1992 no. 205 | MR | Zbl

[14] Branson, T. Q-curvature and spectral invariants, Rend. Circ. Mat. Palermo (2) Suppl. (2005) no. 75, pp. 11-55 | MR

[15] Cheeger, J. Analytic torsion and the heat equation, Ann. of Math. (2), Volume 109 (1979) no. 2, pp. 259-322 | DOI | MR | Zbl

[16] Fried, D. Lefschetz formulas for flows, The Lefschetz centennial conference, Part III, Volume 58 (1987), pp. 19-69 | MR | Zbl

[17] Fried, D. Counting circles, Dynamical systems (College Park, MD, 1986–87) (Lecture Notes in Math), Volume 1342, Springer, Berlin, 1988, pp. 196-215 | MR | Zbl

[18] Fried, D. Torsion and closed geodesics on complex hyperbolic manifolds, Invent. Math., Volume 91 (1988) no. 1, pp. 31-51 | DOI | MR | Zbl

[19] Fuller, F. B. An index of fixed point type for periodic orbits, Amer. J. Math., Volume 89 (1967), pp. 133-148 | DOI | MR | Zbl

[20] Furuta, M.; Steer, B. Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math., Volume 96 (1992) no. 1, pp. 38-102 | DOI | MR | Zbl

[21] Getzler, E. An analogue of Demailly’s inequality for strictly pseudoconvex CR manifolds., J. Differential Geom., Volume 29 (1989) no. 2, pp. 231-244 | MR | Zbl

[22] Gilkey, P. B. Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, 11, Publish or Perish Inc., Wilmington, DE, 1984 | MR | Zbl

[23] Julg, P.; Kasparov, G. Operator K-theory for the group SU (n,1), J. Reine Angew. Math., Volume 463 (1995), pp. 99-152 | MR | Zbl

[24] Kassel, C. Le résidu non commutatif (d’après M. Wodzicki), Astérisque, (177-178):Exp. No. 708, 199–229, 1989. Séminaire Bourbaki, Vol. 1988/89 | Numdam | MR | Zbl

[25] Knudsen, F. F.; Mumford, D. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., Volume 39 (1976) no. 1, pp. 19-55 | MR | Zbl

[26] Milnor, J. W.; Stasheff, J. D. Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press, Princeton, NJ, 1974 | MR | Zbl

[27] Moscovici, H.; Stanton, R. J. R-torsion and zeta functions for locally symmetric manifolds, Invent. Math., Volume 105 (1991) no. 1, pp. 185-216 | DOI | MR | Zbl

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

[29] Nicolaescu, L. I. Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations, Comm. Anal. Geom., Volume 8 (2000) no. 5, pp. 1027-1096 | MR

[30] Ponge, R. Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry, J. Funct. Anal., Volume 252 (2007), pp. 399-463 | DOI | MR

[31] Ponge, R. S. Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, 194, no 906, Mem. Amer. Math. Soc., 2008 | MR

[32] Quillen, D. Determinants of Cauchy-Riemann operators on Riemann surfaces, Funct. Anal. Appl., Volume 214 (1985), pp. 31-34 | DOI | MR | Zbl

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

[34] Rockland, C. Hypoellipticity on the Heisenberg group–representation-theoretic criteria, Trans. Amer. Math. Soc., Volume 240 (1978), pp. 1-52 | DOI | MR | Zbl

[35] Rosenberg, S. The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds, London Mathematical Society Student Texts, 31, Cambridge University Press, Cambridge, 1997 | MR | Zbl

[36] Rumin, M. Un complexe de formes différentielles sur les variétés de contact, C. R. Acad. Sci. Paris Sér. I Math., Volume 310 (1990) no. 6, pp. 101-404 | MR | Zbl

[37] Rumin, M. Formes différentielles sur les variétés de contact, J. Differential Geom., Volume 39 (1994) no. 2, pp. 281-330 | MR | Zbl

[38] Rumin, M. Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal., Volume 10 (2000) no. 2, pp. 407-452 | DOI | MR

[39] Scott, P. The geometries of 3-manifolds, Bull. London Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[40] Seshadri, N. Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds, Bull. Soc. Math. France, Volume 137 (2009) no. 1, pp. 63-91 | Numdam | MR

[41] Stanton, N. K. Spectral invariants of CR manifolds, Michigan Math. J., Volume 36 (1989) no. 2, pp. 267-288 | DOI | MR | Zbl

[42] Tanaka, N. A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9. Kinokuniya Book-Store Co. Ltd., Tokyo, 1975 | MR | Zbl

[43] Tanno, S. Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., Volume 314 (1989) no. 1, pp. 349-379 | DOI | MR | Zbl

[44] Taylor, M. E. Noncommutative microlocal analysis. I, 52, no 313, Mem. Amer. Math. Soc., 1984 | MR | Zbl

[45] Webster, S. M. Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., Volume 13 (1978) no. 1, pp. 25-41 | MR | Zbl

[46] Whittaker, E. T.; Watson, G. N. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Reprinted. Cambridge University Press, New York, 1962 | MR | Zbl

Cité par Sources :