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

Biquard, Olivier; Herzlich, Marc; Rumin, Michel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 40 (2007) no. 4, pp. 589-631.

[1] Apanasov B.V., Geometry and topology of complex hyperbolic and Cauchy-Riemannian manifolds, Russian Math. Surveys 52 (1997) 895-928. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[5] Beals R., Greiner P., Calculus on Heisenberg Manifolds, Annals of Mathematics Studies, vol. 119, Princeton University Press, Princeton, 1988. | MR | Zbl

[6] Beals R., Greiner P.C., Stanton N.K., The heat equation on a CR manifold, J. Diff. Geom. 20 (1984) 343-387. | MR | Zbl

[7] Belgun F., Normal CR structures on compact 3-manifolds, Math. Z. 238 (2001) 441-460. | MR | Zbl

[8] Belgun F., Normal CR structures on $S^3$, Math. Z. 244 (2003) 121-151. | MR | Zbl

[9] Biquard O., Métriques d'Einstein à cusps et équations de Seiberg-Witten, J. reine angew. Math. 490 (1997) 129-154. | Zbl

[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. | Numdam | Zbl

[11] Biquard O., Herzlich M., A Burns-Epstein invariant for ACHE 4-manifolds, Duke Math. J. 126 (2005) 53-100. | Zbl

[12] Bismut J.M., Cheeger J., η-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989) 33-70. | Zbl

[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. | MR | Zbl

[14] Burns D., Epstein C.L., A global invariant for CR three dimensional CR-manifolds, Invent. Math. 92 (1988) 333-348. | MR | Zbl

[15] Burns D., Epstein C.L., Characteristic numbers of bounded domains, Acta Math. 164 (1990) 29-71. | MR | Zbl

[16] Calderbank D.M.J., Singer M.A., Einstein metrics and complex singularities, Invent. Math. 156 (2004) 405-443. | MR | Zbl

[17] Carron G., Pedon E., On the differential form spectrum of hyperbolic manifolds, Ann. Scuol. Norm. Sup. (Pisa) (V) 3 (2004) 707-745. | Numdam | MR

[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. | MR | Zbl

[19] Cheng J.-H., Lee J.M., The Burns-Epstein invariant and deformation of CR structures, Duke Math. J. 60 (1990) 221-254. | Zbl

[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. | MR | Zbl

[21] Chern S.-S., Moser J., Real hypersurfaces in complex manifolds, Acta Math. 133 (1974) 219-271. | MR | Zbl

[22] Chern S.-S., Simons J., Characteristic forms and geometric invariants, Ann. of Math. 99 (1974) 48-69. | MR | Zbl

[23] Dai X., Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc. 4 (1991) 265-321. | MR | Zbl

[24] Getzler E., An analogue of Demailly's inequality for strictly pseudoconvex CR manifolds, J. Diff. Geom. 29 (1989) 231-244. | MR | Zbl

[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.

[26] Graham C.R., Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo Suppl. 63 (2000) 31-42. | MR | Zbl

[27] Harvey F.R., Lawson H.B., On boundaries of complex analytic varieties, Ann. of Math. 102 (1975) 223-290. | MR | Zbl

[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. | MR | Zbl

[29] Hitchin N.J., Einstein metrics and the Eta-invariant, Bolletino U.M.I. 11-B, Suppl., 95-105. | Zbl

[30] Julg P., Kasparov G., Operator K-theory for the group $\mathrm{SU}(n,1)$, J. reine angew. Math. 463 (1995) 99-152. | MR | Zbl

[31] Kamishima Y., Tsuboi T., CR-structures on Seifert manifolds, Invent. Math. 104 (1991) 149-163. | MR | Zbl

[32] Kawasaki T., The Riemann-Roch theorem for complex V-manifolds, Osaka J. Math. 16 (1979) 151-159. | Zbl

[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. | Zbl

[34] Kronheimer P.B., Mrowka T.S., Monopoles and contact structures, Invent. Math. 130 (1997) 209-255. | MR | Zbl

[35] Lee J.M., The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986) 411-429. | MR | Zbl

[36] Lee J.M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math 110 (1988) 157-178. | MR | Zbl

[37] Lisca P., On lens spaces and their symplectic fillings, Math. Res. Lett. 11 (2004) 13-22. | MR | Zbl

[38] Lisca P., Matić G., Tight contact structures and Seiberg-Witten invariants, Invent. Math. 129 (1997) 509-525. | Zbl

[39] Lutz R., Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble) 29 (1979) 283-306. | Numdam | MR | Zbl

[40] Nicolaescu L.I., Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations, Comm. Anal. Geom. 8 (2000) 1027-1096. | Zbl

[41] Ouyang M., Geomeric invariants for Seifert fibred 3-manifolds, Trans. Amer. Math. Soc. 346 (1994) 641-659. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[44] Ponge R., Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, Memoirs AMS, in press, math.AP/0509300.

[45] Rollin Y., Rigidité d'Einstein du plan hyperbolique complexe, J. reine angew. Math. 567 (2004) 175-213. | MR | Zbl

[46] Rumin M., Formes différentielles sur les variétés de contact, J. Diff. Geom. 39 (1994) 281-330. | MR | Zbl

[47] Rumin M., Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal. 10 (2000) 407-452. | MR | Zbl

[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. | MR | Zbl

[49] Seshadri N., Volume renormalisation for complete Einstein-Kähler metrics, Diff. Geom. Appl., in press, math.DG/0404455.

[50] Shubin M.A., Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. | MR | Zbl

[51] Stanton N.K., Spectral invariants of CR manifolds, Michigan Math. J. 36 (1989) 267-288. | MR | Zbl

[52] Stipsicz A.I., On the geography of Stein fillings of certain 3-manifolds, Michigan Math. J. 51 (2003) 327-337. | MR | Zbl

[53] Taylor M.E., Noncommutative Microlocal Analysis. I, Mem. Amer. Math. Soc., vol. 52, 1984. | MR | Zbl

[54] Webster S.M., Pseudo-Hermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978) 25-41. | MR | Zbl

[55] Zhang W.P., Circle bundles, adiabatic limits of η-invariants and Rokhlin congruences, Ann. Inst. Fourier (Grenoble) 44 (1994) 249-270. | Numdam | Zbl