Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3

Olivier Biquard; Marc Herzlich; Michel Rumin

Annales scientifiques de l'École Normale Supérieure (2007)

  • Volume: 40, Issue: 4, page 589-631
  • ISSN: 0012-9593

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Biquard, Olivier, Herzlich, Marc, and Rumin, Michel. "Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3." Annales scientifiques de l'École Normale Supérieure 40.4 (2007): 589-631. <http://eudml.org/doc/82721>.

@article{Biquard2007,
author = {Biquard, Olivier, Herzlich, Marc, Rumin, Michel},
journal = {Annales scientifiques de l'École Normale Supérieure},
language = {eng},
number = {4},
pages = {589-631},
publisher = {Elsevier},
title = {Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3},
url = {http://eudml.org/doc/82721},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Biquard, Olivier
AU - Herzlich, Marc
AU - Rumin, Michel
TI - Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2007
PB - Elsevier
VL - 40
IS - 4
SP - 589
EP - 631
LA - eng
UR - http://eudml.org/doc/82721
ER -

References

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  1. [1] Apanasov B.V., Geometry and topology of complex hyperbolic and Cauchy–Riemannian manifolds, Russian Math. Surveys52 (1997) 895-928. Zbl0920.32024
  2. [2] Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc.77 (1975) 43-69. Zbl0297.58008MR397797
  3. [3] Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry, II, Math. Proc. Cambridge Philos. Soc.78 (1975) 405-432. Zbl0314.58016MR397798
  4. [4] Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry, III, Math. Proc. Cambridge Philos. Soc.79 (1976) 71-99. Zbl0325.58015MR397799
  5. [5] Beals R., Greiner P., Calculus on Heisenberg Manifolds, Annals of Mathematics Studies, vol. 119, Princeton University Press, Princeton, 1988. Zbl0654.58033MR953082
  6. [6] Beals R., Greiner P.C., Stanton N.K., The heat equation on a CR manifold, J. Diff. Geom.20 (1984) 343-387. Zbl0553.58029MR788285
  7. [7] Belgun F., Normal CR structures on compact 3-manifolds, Math. Z.238 (2001) 441-460. Zbl1043.32020MR1869692
  8. [8] Belgun F., Normal CR structures on S 3 , Math. Z.244 (2003) 121-151. Zbl1044.32027MR1981879
  9. [9] Biquard O., Métriques d'Einstein à cusps et équations de Seiberg–Witten, J. reine angew. Math.490 (1997) 129-154. Zbl0891.53029
  10. [10] Biquard O., Métriques d'Einstein asymptotiquement symétriques, Astérisque, vol. 265, Soc. Math. France, Paris, 2000, English translation:, Asymptotically Symmetric Metrics, SMF/AMS Texts and Monographs, vol. 13, AMS, 2006. Zbl0967.53030
  11. [11] Biquard O., Herzlich M., A Burns–Epstein invariant for ACHE 4-manifolds, Duke Math. J.126 (2005) 53-100. Zbl1074.53037
  12. [12] Bismut J.M., Cheeger J., η-invariants and their adiabatic limits, J. Amer. Math. Soc.2 (1989) 33-70. Zbl0671.58037
  13. [13] Bismut J.-M., Freed D.S., The analysis of elliptic families. Dirac operators, eta invariants, and the holonomy theorem, Commun. Math. Phys.107 (1986) 103-163. Zbl0657.58038MR861886
  14. [14] Burns D., Epstein C.L., A global invariant for CR three dimensional CR-manifolds, Invent. Math.92 (1988) 333-348. Zbl0643.32006MR936085
  15. [15] Burns D., Epstein C.L., Characteristic numbers of bounded domains, Acta Math.164 (1990) 29-71. Zbl0704.32005MR1037597
  16. [16] Calderbank D.M.J., Singer M.A., Einstein metrics and complex singularities, Invent. Math.156 (2004) 405-443. Zbl1061.53026MR2052611
  17. [17] Carron G., Pedon E., On the differential form spectrum of hyperbolic manifolds, Ann. Scuol. Norm. Sup. (Pisa) (V)3 (2004) 707-745. Zbl1170.53309MR2124586
  18. [18] Catlin D., Extension of CR structures, in: Several Complex Variables and Complex Geometry, Part 3, Santa Cruz, 1989, Proc. Symp. Pure Math., vol. 52, Amer. Math. Soc., 1991, pp. 27-34. Zbl0790.32020MR1128581
  19. [19] Cheng J.-H., Lee J.M., The Burns–Epstein invariant and deformation of CR structures, Duke Math. J.60 (1990) 221-254. Zbl0704.53028
  20. [20] Cheng S.Y., Yau S.T., On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation, Comm. Pure Appl. Math.33 (1980) 507-544. Zbl0506.53031MR575736
  21. [21] Chern S.-S., Moser J., Real hypersurfaces in complex manifolds, Acta Math.133 (1974) 219-271. Zbl0302.32015MR425155
  22. [22] Chern S.-S., Simons J., Characteristic forms and geometric invariants, Ann. of Math.99 (1974) 48-69. Zbl0283.53036MR353327
  23. [23] Dai X., Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc.4 (1991) 265-321. Zbl0736.58039MR1088332
  24. [24] Getzler E., An analogue of Demailly's inequality for strictly pseudoconvex CR manifolds, J. Diff. Geom.29 (1989) 231-244. Zbl0714.58053MR982172
  25. [25] Gilkey P.B., Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Wilmington, 1984. Zbl0565.58035
  26. [26] Graham C.R., Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo Suppl.63 (2000) 31-42. Zbl0984.53020MR1758076
  27. [27] Harvey F.R., Lawson H.B., On boundaries of complex analytic varieties, Ann. of Math.102 (1975) 223-290. Zbl0317.32017MR425173
  28. [28] Herzlich M., A remark on renormalized volume and Euler characteristic on asymptotically complex hyperbolic Einstein 4-manifolds, Diff. Geom. Appl.25 (2007) 78-91. Zbl1129.53027MR2293643
  29. [29] Hitchin N.J., Einstein metrics and the Eta-invariant, Bolletino U.M.I.11-B, Suppl., 95–105. Zbl0973.53519
  30. [30] Julg P., Kasparov G., Operator K-theory for the group SU ( n , 1 ) , J. reine angew. Math.463 (1995) 99-152. Zbl0819.19004MR1332908
  31. [31] Kamishima Y., Tsuboi T., CR-structures on Seifert manifolds, Invent. Math.104 (1991) 149-163. Zbl0728.32012MR1094049
  32. [32] Kawasaki T., The Riemann–Roch theorem for complex V-manifolds, Osaka J. Math.16 (1979) 151-159. Zbl0405.32010
  33. [33] Komuro M., On Atiyah–Patodi–Singer η-invariant for S 1 -bundles over Riemann surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math.30 (1984) 525-548. Zbl0534.58036
  34. [34] Kronheimer P.B., Mrowka T.S., Monopoles and contact structures, Invent. Math.130 (1997) 209-255. Zbl0892.53015MR1474156
  35. [35] Lee J.M., The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc.296 (1986) 411-429. Zbl0595.32026MR837820
  36. [36] Lee J.M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math110 (1988) 157-178. Zbl0638.32019MR926742
  37. [37] Lisca P., On lens spaces and their symplectic fillings, Math. Res. Lett.11 (2004) 13-22. Zbl1055.57035MR2046195
  38. [38] Lisca P., Matić G., Tight contact structures and Seiberg–Witten invariants, Invent. Math.129 (1997) 509-525. Zbl0882.57008
  39. [39] Lutz R., Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble)29 (1979) 283-306. Zbl0379.53011MR526789
  40. [40] Nicolaescu L.I., Finite energy Seiberg–Witten moduli spaces on 4-manifolds bounding Seifert fibrations, Comm. Anal. Geom.8 (2000) 1027-1096. Zbl1001.58004
  41. [41] Ouyang M., Geomeric invariants for Seifert fibred 3-manifolds, Trans. Amer. Math. Soc.346 (1994) 641-659. Zbl0843.57015MR1257644
  42. [42] Patterson S.J., Perry P.A., The divisor of Selberg's zeta function for Kleinian groups, with an Appendix by C.L. Epstein, Duke Math. J.106 (2001) 321-390. Zbl1012.11083MR1813434
  43. [43] Ponge R., A new short proof of the local index formula and some of its applications, Commun. Math. Phys.241 (2003) 215-234, Erratum, Commun. Math. Phys.248 (2004) 639. Zbl1191.58006MR2013798
  44. [44] Ponge R., Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, Memoirs AMS, in press, math.AP/0509300. Zbl1143.58014
  45. [45] Rollin Y., Rigidité d'Einstein du plan hyperbolique complexe, J. reine angew. Math.567 (2004) 175-213. Zbl1038.53044MR2038308
  46. [46] Rumin M., Formes différentielles sur les variétés de contact, J. Diff. Geom.39 (1994) 281-330. Zbl0973.53524MR1267892
  47. [47] Rumin M., Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal.10 (2000) 407-452. Zbl1008.53033MR1771424
  48. [48] Seeley R.T., Complex powers of an elliptic operator, in: Singular Integrals, Chicago, 1966, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, 1967, pp. 288-307. Zbl0159.15504MR237943
  49. [49] Seshadri N., Volume renormalisation for complete Einstein–Kähler metrics, Diff. Geom. Appl., in press, math.DG/0404455. Zbl1152.32017
  50. [50] Shubin M.A., Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Zbl0616.47040MR883081
  51. [51] Stanton N.K., Spectral invariants of CR manifolds, Michigan Math. J.36 (1989) 267-288. Zbl0685.58033MR1000530
  52. [52] Stipsicz A.I., On the geography of Stein fillings of certain 3-manifolds, Michigan Math. J.51 (2003) 327-337. Zbl1043.53066MR1992949
  53. [53] Taylor M.E., Noncommutative Microlocal Analysis. I, Mem. Amer. Math. Soc., vol. 52, 1984. Zbl0554.35025MR764508
  54. [54] Webster S.M., Pseudo-Hermitian structures on a real hypersurface, J. Diff. Geom.13 (1978) 25-41. Zbl0379.53016MR520599
  55. [55] Zhang W.P., Circle bundles, adiabatic limits of η-invariants and Rokhlin congruences, Ann. Inst. Fourier (Grenoble)44 (1994) 249-270. Zbl0792.57012

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