A survey of the hypoelliptic Laplacian

Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon, Astérisque, no. 322 (2008), p. 39-69

MR 2521653 | Zbl 1180.58001

@incollection{AST_2008__322__39_0,
     author = {Bismut, Jean-Michel},
     title = {A survey of the hypoelliptic Laplacian},
     booktitle = {G\'eom\'etrie diff\'erentielle, physique math\'ematique, math\'ematiques et soci\'et\'e (II) - Volume en l'honneur de Jean-Pierre Bourguignon},
     editor = {Hijazi Oussama},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {322},
     year = {2008},
     pages = {39-69},
     zbl = {1180.58001},
     mrnumber = {2521653},
     language = {en},
     url = {http://www.numdam.org/item/AST_2008__322__39_0}
}

Bismut, Jean-Michel. A survey of the hypoelliptic Laplacian, in Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon, Astérisque, no. 322 (2008), pp. 39-69. http://www.numdam.org/item/AST_2008__322__39_0/

References

[1] A. Berthomieu & J.-M. Bismut - "Quillen metrics and higher analytic torsion forms", J. reine angew. Math. 457 (1994), p. 85-184. | MR 1305280 | Zbl 0804.32017

[2] J.-M. Bismut - "Koszul complexes, harmonic oscillators, and the Todd class", J. Amer. Math. Soc. 3 (1990), p. 159-256. | Article | MR 1017783 | Zbl 0702.58071

[3] J.-M. Bismut, "The hypoelliptic Laplacian on the cotangent bundle", J. Amer. Math. Soc. 18 (2005), p. 379-476. | Article | MR 2137981 | Zbl 1065.35098

[4] J.-M. Bismut, "The hypoelliptic Laplacian and Chern-Gauss-Bonnet", in Differential geometry and physics, Nankai Tracts Math., vol. 10, World Sci. Publ., Hackensack, NJ, 2006, p. 38-52. | Article | MR 2322387 | Zbl 1128.53022

[5] J.-M. Bismut, "The hypoelliptic Dirac operator", in Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, 2008, p. 113-246. | Article | MR 2402405 | Zbl 1157.58011

[6] J.-M. Bismut, "Loop spaces and the hypoelliptic Laplacian", Comm. Pure Appl. Math. 61 (2008), p. 559-593. | Article | MR 2383933 | Zbl 1147.58038

[7] J.-M. Bismut & G. Lebeau - "Complex immersions and Quillen metrics", Publ. Math. I.H.É.S. 74 (1991). | Numdam | MR 1188532 | Zbl 0784.32010

[8] J.-M. Bismut & G. Lebeau, The hypoelliptic Laplacian and Ray-Singer metrics, Annals of Mathematics Studies, vol. AM-167, Princeton University Press, 2008. | MR 2441523 | Zbl 1156.58001

[9] J.-M. Bismut & W. P. Zhang - "An extension of a theorem by Cheeger and Müller", Astérisque 205 (1992), p. 235. | Numdam | MR 1185803 | Zbl 0781.58039

[10] J.-M. Bismut & W. P. Zhang, "Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle", Geom. Funct. Anal. 4 (1994), p. 136-212. | Article | MR 1262703 | Zbl 0830.58030

[11] J. Cheeger - "Analytic torsion and the heat equation", Ann. of Math. 109 (1979), p. 259-322. | Article | MR 528965 | Zbl 0412.58026

[12] B. Helffer & J. Sjöstrand - "Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten", Comm. Partial Differential Equations 10 (1985), p. 245-340. | Article | MR 780068 | Zbl 0597.35024

[13] N. Hitchin - "Harmonic spinors", Advances in Math. 14 (1974), p. 1-55. | MR 358873 | Zbl 0284.58016

[14] L. Hörmander - "Hypoelliptic second order differential equations", Acta Math. 119 (1967), p. 147-171. | Article | MR 222474 | Zbl 0156.10701

[15] A. Kolmogoroff - "Zufällige Bewegungen (zur Theorie der Brownschen Bewegung)", Ann. of Math. 35 (1934), p. 116-117. | Article | MR 1503147 | Zbl 0008.39906

[16] W. Müller - "Analytic torsion and $R$ -torsion of Riemannian manifolds", Adv. in Math. 28 (1978), p. 233-305. | MR 498252 | Zbl 0395.57011

[17] D. Quillen - "Determinants of Cauchy-Riemann operators on Riemann surfaces", Functional Anal. Appl. 19 (1985), p. 31-34. | Article | MR 783704 | Zbl 0603.32016

[18] D. Quillen, "Superconnections and the Chern character", Topology 24 (1985), p. 89-95. | Article | MR 790678 | Zbl 0569.58030

[19] D. B. Ray & I. M. Singer - " $R$ -torsion and the Laplacian on Riemannian manifolds", Advances in Math. 7 (1971), p. 145-210. | MR 295381 | Zbl 0239.58014

[20] E. Witten - "Supersymmetry and Morse theory", J. Differential Geom. 17 (1982), p. 661-692. | Article | MR 683171 | Zbl 0499.53056