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
HTML articles powered by AMS MathViewer

by Jean-Michel Bismut;

J. Amer. Math. Soc. 18 (2005), 379-476

DOI: https://doi.org/10.1090/S0894-0347-05-00479-0

Published electronically: February 28, 2005

PDF | Request permission

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.

References

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
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, DOI 10.1007/978-3-642-58088-8
Alain Berthomieu and Jean-Michel Bismut, Quillen metrics and higher analytic torsion forms, J. Reine Angew. Math. 457 (1994), 85–184. MR 1305280, DOI 10.1515/crll.1994.457.85
Jean-Michel Bismut, Mécanique aléatoire, Lecture Notes in Mathematics, vol. 866, Springer-Verlag, Berlin-New York, 1981 (French). With an English summary. MR 629977, DOI 10.1007/BFb0088591
Jean-Michel Bismut, Index theorem and equivariant cohomology on the loop space, Comm. Math. Phys. 98 (1985), no. 2, 213–237. MR 786574, DOI 10.1007/BF01220509
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, DOI 10.1007/BF01388755
J.-M. Bismut, Bott-Chern currents, excess normal bundles and the Chern character, Geom. Funct. Anal. 2 (1992), no. 3, 285–340. MR 1177315, DOI 10.1007/BF01896875
Jean-Michel Bismut, On certain infinite-dimensional aspects of Arakelov intersection theory, Comm. Math. Phys. 148 (1992), no. 2, 217–248. MR 1178144, DOI 10.1007/BF02100860
Jean-Michel Bismut, Equivariant immersions and Quillen metrics, J. Differential Geom. 41 (1995), no. 1, 53–157. MR 1316553
Jean-Michel Bismut, Holomorphic families of immersions and higher analytic torsion forms, Astérisque 244 (1997), viii+275. MR 1623496
Jean-Michel Bismut, Holomorphic and de Rham torsion, Compos. Math. 140 (2004), no. 5, 1302–1356. MR 2081158, DOI 10.1112/S0010437X04000478
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, DOI 10.1016/j.crma.2004.01.012
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, DOI 10.1016/j.crma.2004.01.009
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, DOI 10.1016/j.crma.2004.01.010

Séminaire: Équations aux Dérivées Partielles, 2003–2004

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, DOI 10.1007/BF01210930
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, DOI 10.1007/BF01238853
Jean-Michel Bismut and Sebastian Goette, Families torsion and Morse functions, Astérisque 275 (2001), x+293 (English, with English and French summaries). MR 1867006
Jean-Michel Bismut and Sebastian Goette, Equivariant de Rham torsions, Ann. of Math. (2) 159 (2004), no. 1, 53–216. MR 2051391, DOI 10.4007/annals.2004.159.53
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

To appear

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, DOI 10.1090/S0894-0347-1995-1303026-5
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
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, DOI 10.1007/BF01895837
Raoul Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR 110104, DOI 10.2307/1970106
Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, DOI 10.2307/1971113
David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 491–517. MR 875085, DOI 10.24033/asens.1515
David Fried, Torsion and closed geodesics on complex hyperbolic manifolds, Invent. Math. 91 (1988), no. 1, 31–51. MR 918235, DOI 10.1007/BF01404911
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, DOI 10.1080/03605308508820379
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
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, DOI 10.7146/math.scand.a-11642
A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116–117 (German). MR 1503147, DOI 10.2307/1968123
Varghese Mathai and Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), no. 1, 85–110. MR 836726, DOI 10.1016/0040-9383(86)90007-8
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, DOI 10.1007/BF01232263
Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, DOI 10.2307/1971113
Daniel Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89–95. MR 790678, DOI 10.1016/0040-9383(85)90047-3
D. Kvillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96 (Russian). MR 783704
D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4

Hamburger Abhandl.

Edward Witten, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), no. 4, 661–692 (1983). MR 683171