The hypoelliptic Laplacian on the cotangent bundle

Bismut, Jean-Michel

doi:10.1090/S0894-0347-05-00479-0

The hypoelliptic Laplacian on the cotangent bundle

Author: Jean-Michel Bismut
Journal: J. Amer. Math. Soc. 18 (2005), 379-476
MSC (2000): Primary 35H10, 58A14, 58J20
DOI: https://doi.org/10.1090/S0894-0347-05-00479-0
Published electronically: February 28, 2005
MathSciNet review: 2137981
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we construct a new version of Hodge theory, where the corresponding Laplacian acts on the total space of the cotangent bundle. This Laplacian is a hypoelliptic operator, which is in general non-self-adjoint. When properly interpreted, it provides an interpolation between classical Hodge theory and the generator of the geodesic flow. The construction is also done in families in the superconnection formalism of Quillen and extends earlier work by Lott and the author.

[A85] M. F. Atiyah, Circular symmetry and stationary-phase approximation, Astérisque 131 (1985), 43–59. Colloquium in honor of Laurent Schwartz, Vol. 1 (Palaiseau, 1983). MR 816738
[BeGeV92] Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720
[BerB94] Alain Berthomieu and Jean-Michel Bismut, Quillen metrics and higher analytic torsion forms, J. Reine Angew. Math. 457 (1994), 85–184. MR 1305280, https://doi.org/10.1515/crll.1994.457.85
[B81] Jean-Michel Bismut, Mécanique aléatoire, Lecture Notes in Mathematics, vol. 866, Springer-Verlag, Berlin-New York, 1981 (French). With an English summary. MR 629977
[B85] Jean-Michel Bismut, Index theorem and equivariant cohomology on the loop space, Comm. Math. Phys. 98 (1985), no. 2, 213–237. MR 786574
[B86] Jean-Michel Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), no. 1, 91–151. MR 813584, https://doi.org/10.1007/BF01388755
[B92a] J.-M. Bismut, Bott-Chern currents, excess normal bundles and the Chern character, Geom. Funct. Anal. 2 (1992), no. 3, 285–340. MR 1177315, https://doi.org/10.1007/BF01896875
[B92b] Jean-Michel Bismut, On certain infinite-dimensional aspects of Arakelov intersection theory, Comm. Math. Phys. 148 (1992), no. 2, 217–248. MR 1178144
[B95] Jean-Michel Bismut, Equivariant immersions and Quillen metrics, J. Differential Geom. 41 (1995), no. 1, 53–157. MR 1316553
[B97] Jean-Michel Bismut, Holomorphic families of immersions and higher analytic torsion forms, Astérisque 244 (1997), viii+275. MR 1623496
[B04a] Jean-Michel Bismut, Holomorphic and de Rham torsion, Compos. Math. 140 (2004), no. 5, 1302–1356. MR 2081158, https://doi.org/10.1112/S0010437X04000478
[B04b] Jean-Michel Bismut, Une déformation de la théorie de Hodge sur le fibré cotangent, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 471–476 (French, with English and French summaries). MR 2057728, https://doi.org/10.1016/j.crma.2004.01.012
[B04c] Jean-Michel Bismut, Le laplacien hypoelliptique sur le fibré cotangent, C. R. Math. Acad. Sci. Paris 338 (2004), no. 7, 555–559 (French, with English and French summaries). MR 2057029, https://doi.org/10.1016/j.crma.2004.01.009
[B04d] Jean-Michel Bismut, Une déformation en famille du complexe de de Rham-Hodge, C. R. Math. Acad. Sci. Paris 338 (2004), no. 8, 623–627 (French, with English and French summaries). MR 2056471, https://doi.org/10.1016/j.crma.2004.01.010
[B04e] J.-M. Bismut.
Le Laplacien hypoelliptique.
In Séminaire: Équations aux Dérivées Partielles, 2003-2004, Sémin. Équ. Dériv. Partielles,Exp. No. XXII, 15. École Polytech., Palaiseau, 2004.
[BF86] Jean-Michel Bismut and Daniel S. Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles, Comm. Math. Phys. 106 (1986), no. 1, 159–176. MR 853982
[BGiSo88] J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys. 115 (1988), no. 1, 49–78. MR 929146
[BG01] Jean-Michel Bismut and Sebastian Goette, Families torsion and Morse functions, Astérisque 275 (2001), x+293 (English, with English and French summaries). MR 1867006
[BG04] Jean-Michel Bismut and Sebastian Goette, Equivariant de Rham torsions, Ann. of Math. (2) 159 (2004), no. 1, 53–216. MR 2051391, https://doi.org/10.4007/annals.2004.159.53
[BL91] Jean-Michel Bismut and Gilles Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. 74 (1991), ii+298 pp. (1992). MR 1188532
[BL05] J.-M. Bismut and G. Lebeau.
The hypoelliptic Laplacian and Ray-Singer metrics.
To appear, 2005.
[BLo95] Jean-Michel Bismut and John Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), no. 2, 291–363 (English, with English and French summaries). MR 1303026, https://doi.org/10.1090/S0894-0347-1995-1303026-5
[BZ92] Jean-Michel Bismut and Weiping Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992), 235 (English, with French summary). With an appendix by François Laudenbach. MR 1185803
[BZ94] J.-M. Bismut and W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal. 4 (1994), no. 2, 136–212 (English, with English and French summaries). MR 1262703, https://doi.org/10.1007/BF01895837
[Bo59] Raoul Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR 110104, https://doi.org/10.2307/1970106
[C79] Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, https://doi.org/10.2307/1971113
[Fr86] David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 491–517. MR 875085
[Fr88] David Fried, Torsion and closed geodesics on complex hyperbolic manifolds, Invent. Math. 91 (1988), no. 1, 31–51. MR 918235, https://doi.org/10.1007/BF01404911
[HSj85] B. Helffer and J. Sjöstrand, Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations 10 (1985), no. 3, 245–340 (French). MR 780068, https://doi.org/10.1080/03605308508820379
[Hö67] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, https://doi.org/10.1007/BF02392081
[KM76] Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55. MR 437541, https://doi.org/10.7146/math.scand.a-11642
[Ko34] A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116–117 (German). MR 1503147, https://doi.org/10.2307/1968123
[MaQ86] Varghese Mathai and Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), no. 1, 85–110. MR 836726, https://doi.org/10.1016/0040-9383(86)90007-8
[MoSt91] Henri Moscovici and Robert J. Stanton, 𝑅-torsion and zeta functions for locally symmetric manifolds, Invent. Math. 105 (1991), no. 1, 185–216. MR 1109626, https://doi.org/10.1007/BF01232263
[Mü78] Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, https://doi.org/10.2307/1971113
[Q85a] Daniel Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89–95. MR 790678, https://doi.org/10.1016/0040-9383(85)90047-3
[Q85b] D. Kvillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96 (Russian). MR 783704
[RS71] D. B. Ray and I. M. Singer, 𝑅-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, https://doi.org/10.1016/0001-8708(71)90045-4
[Re35] K. Reidemeister.
Homotopieringe und Linsenraüm.
Hamburger Abhandl., pages 102-109, 1935.
[W82] Edward Witten, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), no. 4, 661–692 (1983). MR 683171