Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

Journal of Functional Analysis

Volumn 252, Issue 2, 2007, Pages 399-463

  Ponge, Raphaël ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

Heisenberg calculus; Hypoelliptic operators; Noncommutative geometry; Noncommutative residue

Indexed keywords

EID: 35248899208     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2007.07.001     Document Type: Article

References (70)

1. Beals R., and Greiner P.C. Calculus on Heisenberg Manifolds. Ann. of Math. Stud. vol. 119 (1988), Princeton Univ. Press, Princeton, NJ
2. Beals R., Greiner P.C., and Stanton N.K. The heat equation on a CR manifold. J. Differential Geom. 20 2 (1984) 343-387
3. Bellaïche A. The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry. Progr. Math. vol. 144 (1996), Birkhäuser, Basel 1-78
4. Berline N., Getzler E., and Vergne M. Heat Kernels and Dirac Operators. Grundlehren Math. Wiss. vol. 298 (1992), Springer, Berlin
5. Biquard O. Métriques d'Einstein asymptotiquement symétriques. Astérisque 265 (2000)
6. Bony J.M. Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier 19 (1969) 277-304
7. Boutet de Monvel L. Hypoelliptic operators with double characteristics and related pseudo-differential operators. Comm. Pure Appl. Math. 27 (1974) 585-639
8. Boutet de Monvel L., and Guillemin V. The Spectral Theory of Toeplitz Operators. Ann. of Math. Stud. vol. 99 (1981), Princeton Univ. Press, Princeton, NJ
9. Christ M., Geller D., Głowacki P., and Polin L. Pseudodifferential operators on groups with dilations. Duke Math. J. 68 (1992) 31-65
10. Connes A. The action functional in noncommutative geometry. Comm. Math. Phys. 117 4 (1988) 673-683
11. Connes A. Noncommutative Geometry (1994), Academic Press, San Diego, CA
12. Connes A. Gravity coupled with matter and the foundation of non-commutative geometry. Comm. Math. Phys. 182 1 (1996) 155-176
13. Connes A., and Moscovici H. The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 2 (1995) 174-243
14. Dixmier J. Existence de traces non normales. C. R. Acad. Sci. Paris Sér. A-B 262 (1966) A1107-A1108
15. Dynin A. Pseudodifferential operators on the Heisenberg group. Dokl. Akad. Nauk SSSR 225 (1975) 1245-1248
16. Dynin A. An algebra of pseudodifferential operators on the Heisenberg groups. Symbolic calculus. Dokl. Akad. Nauk SSSR 227 (1976) 792-795
17. Eliashberg Y., and Thurston W. Confoliations. Univ. Lecture Ser. vol. 13 (1998), Amer. Math. Soc., Providence, RI
18. Epstein C.L., and Melrose R.B. The Heisenberg algebra, index theory and homology. preprint, 1998, available at. http://www-math.mit.edu/~rbm
19. Epstein C.L., Melrose R.B., and Mendoza G. Resolvent of the Laplacian on strictly pseudoconvex domains. Acta Math. 167 1-2 (1991) 1-106
20. Fedosov B.V., Golse F., Leichtnam E., and Schrohe E. The noncommutative residue for manifolds with boundary. J. Funct. Anal. 142 1 (1996) 1-31
21. Fefferman C. Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. of Math. (2) 103 2 (1976) 395-416
22. Fefferman C.L., and Sánchez-Calle A. Fundamental solutions for second order subelliptic operators. Ann. of Math. (2) 124 2 (1986) 247-272
23. b complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974) 429-522
24. Gilkey P.B. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Math. Lecture Ser. vol. 11 (1984), Publish or Perish, Wilmington, DE
25. Gohberg I.C., and Kreǐn M.G. Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. Math. Monogr. vol. 18 (1969), Amer. Math. Soc., Providence, RI
26. Greenleaf A. The first eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold. Comm. Partial Differential Equations 10 2 (1985) 191-217
27. Gromov M. Carnot-Carathéodory spaces seen from within. Sub-Riemannian Geometry. Progr. Math. vol. 144 (1996), Birkhäuser, Basel 79-323
28. Guillemin V. A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. Math. 55 2 (1985) 131-160
29. Guillemin V. Gauged Lagrangian distributions. Adv. Math. 102 2 (1993) 184-201
30. Guillemin V. Residue traces for certain algebras of Fourier integral operators. J. Funct. Anal. 115 2 (1993) 391-417
31. Helffer B., and Nourrigat J. Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs. Progr. Math. vol. 58 (1986), Birkhäuser, Boston, MA
32. Jerison D., and Lee J.M. The Yamabe problem on CR manifolds. J. Differential Geom. 25 2 (1987) 167-197
33. Julg P., and Kasparov G. Operator K-theory for the group SU (n, 1). J. Reine Angew. Math. 463 (1995) 99-152
34. Kalau W., and Walze M. Gravity, non-commutative geometry and the Wodzicki residue. J. Geom. Phys. 16 4 (1995) 327-344
35. Kassel C. Le résidu non commutatif (d'après M. Wodzicki). Séminaire Bourbaki, vol. 1988/1989. Astérisque 41 708 (1989) 177-178 199-229
36. Kastler D. The Dirac operator and gravitation. Comm. Math. Phys. 166 3 (1995) 633-643
37. Kohn J.J. Boundaries of complex manifolds. Proc. Conf. Complex Analysis. Minneapolis, MN, 1964 (1965), Springer, Berlin 81-94
38. Kohn J.J., and Rossi H. On the extension of holomorphic functions from the boundary of a complex manifold. Ann. of Math. 81 (1965) 451-472
39. Kontsevich M., and Vishik S. Geometry of determinants of elliptic operators. Functional Analysis on the Eve of the 21st Century, vol. 1. New Brunswick, NJ, 1993. Progr. Math. vol. 131 (1995), Birkhäuser Boston, Boston, MA 173-197
40. Lee J.M. The Fefferman metric and pseudo-Hermitian invariants. Trans. Amer. Math. Soc. 296 1 (1986) 411-429
41. Lesch M. On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols. Ann. Global Anal. Geom. 17 2 (1999) 151-187
42. Machedon M. Estimates for the parametrix of the Kohn Laplacian on certain domains. Invent. Math. 91 2 (1988) 339-364
43. Mathai V., Melrose R., and Singer I. Fractional index theory. J. Differential Geom. 74 (2006) 265-292
44. R. Melrose, V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, preprint, arXiv, June 1996
45. Nagel A., Stein E.M., and Wainger S. Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155 1-2 (1985) 103-147
46. Paycha S., and Rosenberg S. Curvature on determinant bundles and first Chern forms. J. Geom. Phys. 45 3-4 (2003) 393-429
47. Ponge R. Calcul fonctionnel sous-elliptique et résidu non commutatif sur les variétés de Heisenberg. C. R. Acad. Sci. Paris, Sér. I 332 (2001) 611-614
48. Ponge R. Géométrie spectrale et formules d'indices locales pour les variétés CR et contact. C. R. Acad. Sci. Paris, Sér. I 332 (2001) 735-738
49. Ponge R. Spectral asymmetry, zeta functions and the noncommutative residue. Internat. J. Math. 17 (2006) 1065-1090
50. Ponge R. The tangent groupoid of a Heisenberg manifold. Pacific Math. J. 227 (2006) 151-175
51. R. Ponge, Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, e-print, arXiv, September 2005, 140 pp. Mem. Amer. Math. Soc., in press
52. R. Ponge, Noncommutative residue invariants for CR and contact manifolds, e-print, arXiv, October 2005, 30 pp., J. Reine Angew. Math., in press
53. R. Ponge, Noncommutative geometry and lower dimensional volumes in Riemannian geometry, e-print, arXiv, July 2007
54. R. Ponge, Hypoelliptic functional calculus on Heisenberg manifolds. A resolvent approach, e-print, arXiv, September 2007
55. Rockland C. Hypoellipticity on the Heisenberg group-representation-theoretic criteria. Trans. Amer. Math. Soc. 240 (1978) 1-52
56. Rockland C. Intrinsic nilpotent approximation. Acta Appl. Math. 8 3 (1987) 213-270
57. Rothschild L., and Stein E. Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976) 247-320
58. Rumin M. Formes différentielles sur les variétés de contact. J. Differential Geom. 39 2 (1994) 281-330
59. Sánchez-Calle A. Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78 1 (1984) 143-160
60. Schrohe E. Noncommutative residues and manifolds with conical singularities. J. Funct. Anal. 150 1 (1997) 146-174
61. Stanton N.K. Spectral invariants of CR manifolds. Michigan Math. J. 36 2 (1989) 267-288
62. Tanaka N. A Differential Geometric Study on Strongly Pseudo-convex Manifolds. Lectures in Math., Department of Math., Kyoto Univ. vol. 9 (1975), Kinokuniya, Tokyo
63. Taylor M.E. Noncommutative microlocal analysis. I. Mem. Amer. Math. Soc. 52 313 (1984)
64. E. Van Erp, PhD thesis, Pennsylvania State Univ., 2005
65. S. Vassout, Feuilletages et résidu non commutatif longitudinal, PhD thesis, Univ. Paris 7, 2001
66. Webster S. Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom. 13 1 (1978) 25-41
67. Wodzicki M. Local invariants of spectral asymmetry. Invent. Math. 75 1 (1984) 143-177
68. M. Wodzicki, Spectral asymmetry and noncommutative residue, Habilitation thesis, Steklov Institute, (former) Soviet Academy of Sciences, Moscow, 1984 (in Russian)
69. Wodzicki M. Noncommutative residue. I. Fundamentals. K-Theory, Arithmetic and Geometry. Moscow, 1984-1986. Lecture Notes in Math. vol. 1289 (1987), Springer, Berlin 320-399
70. Wodzicki M. The long exact sequence in cyclic homology associated with extensions of algebras. C. R. Acad. Sci. Paris Sér. I Math. 306 9 (1988) 399-403