Le Laplacien hypoelliptique

Bismut, Jean-Michel

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 21, 15 p.

Résumé
Abstract

On construit une nouvelle théorie de Hodge sur le fibré cotangent d’une variété Riemannienne $X$ . Le Laplacien correspondant est un opérateur hypoelliptique d’ordre deux, qui est autoadjoint relativement à une forme Hermitienne de signature $(\infty, \infty)$ . Cette théorie de Hodge interpole entre la théorie de Hodge habituelle sur $X$ et le flot géodésique sur $T * X$ .

We construct a new Hodge theory on the cotangent bundle of a Riemannian manifold $X$ . The corresponding Laplacian is a second order hypoelliptic operator, which is self-adjoint with respect to a Hermitian form whose signature is $(\infty, \infty)$ . This Hodge theory interpolates between the classical Hodge theory on $X$ and the geodesic flow on $T * X$ .

MR 2117053 | 3 citations dans Numdam

BibTeX
RIS

@article{SEDP_2003-2004____A21_0,
     author = {Bismut, Jean-Michel},
     title = {Le {Laplacien} hypoelliptique},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:21},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2003-2004},
     mrnumber = {2117053},
     language = {fr},
     url = {http://www.numdam.org/item/SEDP_2003-2004____A21_0/}
}

TY  - JOUR
AU  - Bismut, Jean-Michel
TI  - Le Laplacien hypoelliptique
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:21
PY  - 2003-2004
DA  - 2003-2004///
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2003-2004____A21_0/
UR  - https://www.ams.org/mathscinet-getitem?mr=2117053
LA  - fr
ID  - SEDP_2003-2004____A21_0
ER  -

Bismut, Jean-Michel. Le Laplacien hypoelliptique. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 21, 15 p. http://www.numdam.org/item/SEDP_2003-2004____A21_0/

Bibliographie
Cité par

[A85] M. F. Atiyah. Circular symmetry and stationary-phase approximation. Astérisque, (131) :43–59, 1985. Colloquium in honor of Laurent Schwartz, Vol. 1 (Palaiseau, 1983). | MR 816738 | Zbl 0578.58039

[BeGV92] N. Berline, E. Getzler, M. Vergne. Heat kernels and Dirac operators. Grundl. Math. Wiss. Band 298. Springer-Verlag, Berlin, 1992. | MR 1215720 | Zbl 0744.58001

[B04a] J.-M. Bismut. The hypoelliptic Laplacian on the cotangent bundle. À paraître, 2004. | MR 2057029

[B04b] J.-M. Bismut. Une déformation de la théorie de Hodge sur le fibré cotangent. C. R. Acad. Sci. Paris Sér. I, 338 :471–476, 2004. | MR 2057728 | Zbl 1044.58005

[B04c] J.-M. Bismut. Le Laplacien hypoelliptique sur le fibré cotangent. C. R. Math. Acad. Sci. Paris Sér. I, 338 :555–559, 2004. | MR 2057029 | Zbl 1049.58005

[B04d] J.-M. Bismut. Une déformation en famille du complexe de de Rham-Hodge. C. R. Math. Acad. Sci. Paris Sér. I, 338 :623–627, 2004. | MR 2056471 | Zbl 1058.58009

[BL91] J.-M. Bismut, G. Lebeau. Complex immersions and Quillen metrics. Inst. Hautes Études Sci. Publ. Math., (74) :ii+298 pp. (1992), 1991. | Numdam | MR 1188532 | Zbl 0784.32010

[BL04] J.-M. Bismut, G. Lebeau. The hypoelliptic Laplacian and Ray-Singer metrics. À paraître, 2004.

[BZ92] J.-M. Bismut, W. Zhang. An extension of a theorem by Cheeger and Müller. Astérisque, (205) :235, 1992. With an appendix by François Laudenbach. | MR 1185803 | Zbl 0781.58039

[BZ94] J.-M. Bismut, W. Zhang. Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle. Geom. Funct. Anal., 4(2) :136–212, 1994. | MR 1262703 | Zbl 0830.58030

[Bo59] R. Bott. The stable homotopy of the classical groups. Ann. of Math. (2), 70 :313–337, 1959. | MR 110104 | Zbl 0129.15601

[C79] J. Cheeger. Analytic torsion and the heat equation. Ann. of Math. (2), 109(2) :259–322, 1979. | MR 528965 | Zbl 0412.58026

[Ch44] S.S. Chern. A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. of Math. (2), 45 :747–752, 1944. | MR 11027 | Zbl 0060.38103

[HS85] B. Helffer, J. Sjöstrand. Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten. Comm. Partial Differential Equations, 10(3) :245–340, 1985. | MR 780068 | Zbl 0597.35024

[Hö67] L. Hörmander. Hypoelliptic second order differential equations. Acta Math., 119 :147–171, 1967. | MR 222474 | Zbl 0156.10701

[MQ86] V. Mathai, D. Quillen. Superconnections, Thom classes, and equivariant differential forms. Topology, 25(1) :85–110, 1986. | MR 836726 | Zbl 0592.55015

[McS67] H. P. McKean, Jr., I. M. Singer. Curvature and the eigenvalues of the Laplacian. J. Differential Geometry, 1(1) :43–69, 1967. | MR 217739 | Zbl 0198.44301

[Mi65] J. Milnor. Lectures on the $h$ -cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, N.J., 1965. | MR 190942 | Zbl 0161.20302

[Mü78] W. Müller. Analytic torsion and $R$ -torsion of Riemannian manifolds. Adv. in Math., 28(3) :233–305, 1978. | MR 498252 | Zbl 0395.57011

[RS71] D. B. Ray, I. M. Singer. $R$ -torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7 :145–210, 1971. | MR 295381 | Zbl 0239.58014

[Rei35] K. Reidemeister. Homotopieringe und Linsenraüm. Hamburger Abhandl., pages 102–109, 1935. | Zbl 0011.32404

[Sm61] S. Smale. On gradient dynamical systems. Ann. of Math. (2), 74 :199–206, 1961. | MR 133139 | Zbl 0136.43702

[Sm67] S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 73 :747–817, 1967. | MR 228014 | Zbl 0202.55202

[T49] R. Thom. Sur une partition en cellules associée à une fonction sur une variété. C. R. Acad. Sci. Paris, 228 :973–975, 1949. | MR 29160 | Zbl 0034.20802

[W82] E. Witten. Supersymmetry and Morse theory. J. Differential Geom., 17(4) :661–692 (1983), 1982. | MR 683171 | Zbl 0499.53056