Analytic torsions on contact manifolds
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, p. 727-782

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.

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.

DOI : https://doi.org/10.5802/aif.2693
Classification:  58J52,  32V05,  32V20,  11M36,  37C30
Keywords: analytic torsion, contact complex, CR Seifert manifold, trace formula
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     pages = {727-782},
     doi = {10.5802/aif.2693},
     mrnumber = {2985515},
     zbl = {1264.58027},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_2_727_0}
}
Rumin, Michel; Seshadri, Neil. Analytic torsions on contact manifolds. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 727-782. doi : 10.5802/aif.2693. http://www.numdam.org/item/AIF_2012__62_2_727_0/

[1] Bär, C.; Moroianu, S. Heat kernel asymptotics for roots of generalized Laplacians, Internat. J. Math., Tome 14 (2003) no. 4, pp. 397-412 | Article | MR 1984660

[2] Baston, R. J.; Eastwood, M. G. The Penrose transform: Its interaction with representation theory, The Clarendon Press, Oxford University Press, New York, Oxford Mathematical Monographs (1989) | MR 1038279 | Zbl 0726.58004

[3] Beals, R.; Greiner, P. Calculus on Heisenberg manifolds, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 119 (1988) | MR 953082 | Zbl 0654.58033

[4] Beals, R.; Greiner, P. C.; Stanton, N. K. The heat equation and geometry of CR manifolds, Bull. Amer. Math. Soc. (N.S.), Tome 10 (1984) no. 2, p. 275-276 | Article | MR 733694 | Zbl 0543.58024

[5] Belgun, F. A. Normal CR structures on S 3 , Math. Z., Tome 244 (2003) no. 1, pp. 123-151 | Article | MR 1981879

[6] Biquard, O.; Herzlich, M. Burns-Epstein invariant for ACHE 4-manifolds, Duke Math. J., Tome 126 (2005) no. 1, pp. 53-100 | Article | MR 2110628

[7] Biquard, O.; Herzlich, M.; Rumin, M. Diabatic limit, eta invariants and Cauchy-Riemann manifolds of dimension 3, Ann. Sci. Ecole Norm. Sup. (4), Tome 40 (2007) no. 4, pp. 589-631 | MR 2191527

[8] Bismut, J.-M. The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., Tome 18 (2005) no. 2, p. 379-476 (electronic) | Article | MR 2137981

[9] Bismut, J.-M. Loop spaces and the hypoelliptic Laplacian, Comm. Pure Appl. Math., Tome 61 (2008) no. 4, pp. 559-593 | Article | MR 2383933

[10] Bismut, J.-M.; Gillet, H.; C., Soulé Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys., Tome 115 (1988) no. 1, pp. 49-78 | Article | MR 929146 | Zbl 0651.32017

[11] Bismut, J.-M.; Gillet, H.; C., Soulé Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Comm. Math. Phys., Tome 115 (1988) no. 2, pp. 301-351 | Article | MR 931666 | Zbl 0651.32017

[12] Bismut, J.-M.; Lebeau, G. The hypoelliptic Laplacian and Ray-Singer metrics, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 167 (2008) | MR 2441523

[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 1185803 | Zbl 0781.58039

[14] Branson, T. Q-curvature and spectral invariants, Rend. Circ. Mat. Palermo (2) Suppl. (2005) no. 75, pp. 11-55 | MR 2152355

[15] Cheeger, J. Analytic torsion and the heat equation, Ann. of Math. (2), Tome 109 (1979) no. 2, pp. 259-322 | Article | MR 528965 | Zbl 0412.58026

[16] Fried, D. Lefschetz formulas for flows, The Lefschetz centennial conference, Part III, Amer. Math. Soc., Providence, RI, Mexico City, 1984, Tome 58 (1987), pp. 19-69 | MR 893856 | Zbl 0619.58034

[17] Fried, D. Counting circles, Dynamical systems (College Park, MD, 1986–87), Springer, Berlin (Lecture Notes in Math) Tome 1342 (1988), pp. 196-215 | MR 970556 | Zbl 0662.58033

[18] Fried, D. Torsion and closed geodesics on complex hyperbolic manifolds, Invent. Math., Tome 91 (1988) no. 1, pp. 31-51 | Article | MR 918235 | Zbl 0658.53061

[19] Fuller, F. B. An index of fixed point type for periodic orbits, Amer. J. Math., Tome 89 (1967), pp. 133-148 | Article | MR 209600 | Zbl 0152.40204

[20] Furuta, M.; Steer, B. Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math., Tome 96 (1992) no. 1, pp. 38-102 | Article | MR 1185787 | Zbl 0769.58009

[21] Getzler, E. An analogue of Demailly’s inequality for strictly pseudoconvex CR manifolds., J. Differential Geom., Tome 29 (1989) no. 2, pp. 231-244 | MR 982172 | Zbl 0714.58053

[22] Gilkey, P. B. Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Publish or Perish Inc., Wilmington, DE, Mathematics Lecture Series, Tome 11 (1984) | MR 783634 | Zbl 0565.58035

[23] Julg, P.; Kasparov, G. Operator K-theory for the group SU (n,1), J. Reine Angew. Math., Tome 463 (1995), pp. 99-152 | MR 1332908 | Zbl 0819.19004

[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 1040574 | Zbl 0701.58054

[25] Knudsen, F. F.; Mumford, D. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., Tome 39 (1976) no. 1, pp. 19-55 | MR 437541 | Zbl 0343.14008

[26] Milnor, J. W.; Stasheff, J. D. Characteristic classes, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 76 (1974) | MR 440554 | Zbl 0298.57008

[27] Moscovici, H.; Stanton, R. J. R-torsion and zeta functions for locally symmetric manifolds, Invent. Math., Tome 105 (1991) no. 1, pp. 185-216 | Article | MR 1109626 | Zbl 0789.58073

[28] Müller, W. Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., Tome 28 (1978) no. 3, pp. 233-305 | Article | MR 498252 | Zbl 0395.57011

[29] Nicolaescu, L. I. Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations, Comm. Anal. Geom., Tome 8 (2000) no. 5, pp. 1027-1096 | MR 1846125

[30] Ponge, R. Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry, J. Funct. Anal., Tome 252 (2007), pp. 399-463 | Article | MR 2360923

[31] Ponge, R. S. Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, Mem. Amer. Math. Soc. Tome 194, no 906 (2008) | MR 2417549

[32] Quillen, D. Determinants of Cauchy-Riemann operators on Riemann surfaces, Funct. Anal. Appl., Tome 214 (1985), pp. 31-34 | Article | MR 783704 | Zbl 0603.32016

[33] Ray, D. B.; Singer, I. M. R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Tome 7 (1971), pp. 145-210 | Article | MR 295381 | Zbl 0239.58014

[34] Rockland, C. Hypoellipticity on the Heisenberg group–representation-theoretic criteria, Trans. Amer. Math. Soc., Tome 240 (1978), pp. 1-52 | Article | MR 486314 | Zbl 0326.22007

[35] Rosenberg, S. The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds, Cambridge University Press, Cambridge, London Mathematical Society Student Texts, Tome 31 (1997) | MR 1462892 | Zbl 0868.58074

[36] Rumin, M. Un complexe de formes différentielles sur les variétés de contact, C. R. Acad. Sci. Paris Sér. I Math., Tome 310 (1990) no. 6, pp. 101-404 | MR 1046521 | Zbl 0694.57010

[37] Rumin, M. Formes différentielles sur les variétés de contact, J. Differential Geom., Tome 39 (1994) no. 2, pp. 281-330 | MR 1267892 | Zbl 0973.53524

[38] Rumin, M. Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal., Tome 10 (2000) no. 2, pp. 407-452 | Article | MR 1771424

[39] Scott, P. The geometries of 3-manifolds, Bull. London Math. Soc., Tome 15 (1983) no. 5, pp. 401-487 | Article | MR 705527 | Zbl 0561.57001

[40] Seshadri, N. Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds, Bull. Soc. Math. France, Tome 137 (2009) no. 1, pp. 63-91 | Numdam | MR 2496701

[41] Stanton, N. K. Spectral invariants of CR manifolds, Michigan Math. J., Tome 36 (1989) no. 2, pp. 267-288 | Article | MR 1000530 | Zbl 0685.58033

[42] Tanaka, N. A differential geometric study on strongly pseudo-convex manifolds, Department of Mathematics, Kyoto University, No. 9. Kinokuniya Book-Store Co. Ltd., Tokyo, Lectures in Mathematics (1975) | MR 399517 | Zbl 0331.53025

[43] Tanno, S. Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., Tome 314 (1989) no. 1, pp. 349-379 | Article | MR 1000553 | Zbl 0677.53043

[44] Taylor, M. E. Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc. Tome 52, no 313 (1984) | MR 764508 | Zbl 0554.35025

[45] Webster, S. M. Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., Tome 13 (1978) no. 1, pp. 25-41 | MR 520599 | Zbl 0379.53016

[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 178117 | Zbl 0105.26901