On the differential form spectrum of hyperbolic manifolds

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1. [Alb] P. Albuquerque, Patterson-Sullivan theory in higher rank symmetric spaces, Geom. Funct. Anal. 9 (1999), 1-28. Zbl0954.53031MR1675889
2. [AAR] G. E. Andrews – R. Askey – R. Roy, “Special functions”, Encyclopedia Math. Appl. 71., Cambridge Univ. Press, 1999. Zbl0920.33001MR1688958
3. [ADY] J.-Ph. Anker – E. Damek – C. Yacoub, Spherical analysis on harmonic $AN$ groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 643-679. Zbl0881.22008MR1469569
4. [AJ] J.-Ph. Anker – L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035-1091. Zbl0942.43005MR1736928
5. [BCG1] G. Besson – G. Courtois – S. Gallot, Lemme de Schwarz réel et applications géométriques, Acta Math. 183 (1999), 145-169. Zbl1035.53038MR1738042
6. [BCG2] G. Besson – G. Courtois – S. Gallot, Hyperbolic manifolds, almagamated products and critical exponents, C. R. Math. Acad. Sci. Paris 336 (2003), 257-261. Zbl1026.57013MR1968269
7. [Bor] A. Borel, The $L^2$-cohomology of negatively curved Riemannian symmetric spaces, Ann. Acad. Scient. Fenn. Series A.I. Math. 10 (1985), 95-105. Zbl0586.57022MR802471
8. [BW] A. Borel – N. R. Wallach, “Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups", second ed”., Math. Surveys Monogr. 67, Amer. Math. Soc., 2000. Zbl0980.22015MR1721403
9. [Bow] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-25. Zbl0439.30032MR556580
10. [BOS] T. P. Branson – G. Ólafsson – H. Schlichtkrull, A bundle-valued Radon transform, with applications to invariant wave equations, Quart. J. Math. Oxford (2) 45 (1994), 429-461. Zbl0924.43010MR1315457
11. [BO1] U. Bunke – M. Olbrich, The spectrum of Kleinian manifolds, J. Funct. Anal. 172 (2000), 76-164. Zbl0966.43004MR1749869
12. [BO2] U. Bunke – M. Olbrich, Nonexistence of invariant distributions supported on the limit set, The 2000 Twente Conference on Lie groups (Enschede), Acta Appl. Math. 73 (2002), 3-13. Zbl1018.22008MR1926489
13. [BO3] U. Bunke – M. Olbrich, Towards the trace formula for convex-cocompact groups, preprint, 2000. MR1749869
14. [CGH] D. M. J. Calderbank – P. Gauduchon – M. Herzlich, Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal. 173 (2000), 214-255. Zbl0960.58010MR1760284
15. [CSZ] H. D. Cao – Y. Shen – S. Zhu, The structure of stable minimal hypersurfaces in ${ℝ}^{n+1}$, Math. Res. Lett. 4 (1997), 637-644. Zbl0906.53004MR1484695
16. [Car] G. Carron, Une suite exacte en $L^2$-cohomologie, Duke Math. J. 95 (1998), 343-372. Zbl0951.58024MR1652017
17. [CG] J. Cheeger – D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971-1972), 119-128. Zbl0223.53033MR303460
18. [Col] Y. Colin de Verdière, Théorie spectrale des surfaces de Riemann d’aire infinie, In: “Colloque en l’honneur de Laurent Schwartz” (Palaiseau, 1983), Vol. 2, Astérisque 132, Soc. Math. France, 1985, pp. 259-275. Zbl0582.58030MR816771
19. [CM] A. Connes – H. Moscovici, The $L^2$-index theorem for homogeneous spaces of Lie groups, Ann. of Math. (2) 115 (1982), 292-330. Zbl0515.58031MR647808
20. [Cor] K. Corlette, Hausdorff dimensions of limit sets, Invent. Math. 102 (1990), 521-541. Zbl0744.53030MR1074486
21. [Don] H. Donnelly, The differential form spectrum of hyperbolic space, Manuscripta Math. 33 (1981), 365-385. Zbl0464.58020MR612619
22. [DF] H. Donnelly – Ch. Fefferman, $L^2$-cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), 593-618. Zbl0532.58027MR727705
23. [Els] J. Elstrodt, Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, Teil I. Math. Ann. 203 (1973), 295-330; Teil II. Math. Z. 132 (1973), 99-134; Teil III. Math. Ann. 208 (1974), 99-132. Zbl0265.10017MR360472
24. [EMM] Ch. Epstein – R. Melrose – G. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1-106. Zbl0758.32010MR1111745
25. [Far] J. Faraut, Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, In: “Analyse harmonique”, J. L. Clerc et. al. (eds), Les cours du C.I.M.P.A., Nice, 1982, pp. 315-446. 
26. [GM] S. Gallot – D. Meyer, Opérateur de courbure et Laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), 259-284. Zbl0316.53036MR454884
27. [HV] P. de la Harpe – A. Valette, La propriété $\left(T\right)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 Soc. Math. France, (1989). Zbl0759.22001
28. [Heb] E. Hebey, “Sobolev spaces on Riemannian manifolds”, Lecture Notes in Math., 1635, Springer, 1996. Zbl0866.58068MR1481970
29. [Hel] S. Helgason, “Differential Geometry, Lie groups, and Symmetric Spaces”, Academic Press, 1978. Zbl0451.53038MR514561
30. [Ize] H. Izeki, Limit sets of Kleinian groups and conformally flat Riemannian manifolds, Invent. Math. 122 (1995), 603-625. Zbl0854.53035MR1359605
31. [IN] H. Izeki – S. Nayatani, Canonical metric on the domain of discontinuity of a Kleinian group, Sémin. Théor. Spectr. Géom. (Univ. Grenoble-I) 16 (1997-1998), 9-32. Zbl0979.53036MR1666506
32. [Kna] A. W. Knapp, “Representation Theory of Semisimple Groups. An Overview Based on Examples”, Princeton Math. Ser. 36, Princeton Univ. Press, 1986. Zbl0604.22001MR855239
33. [KR] J. Kohn – H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451-472. Zbl0166.33802MR177135
34. [Leu] E. Leuzinger, Critical exponents of discrete groups and $L^2$-spectrum, Proc. Amer. Math. Soc. 132 (2004), 919-927. Zbl1034.22009MR2019974
35. [Li] P. Li, On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math. 99 (1990), 579-600. Zbl0695.53052MR1032881
36. [LiR] P. Li – M. Ramachandran, Kähler manifolds with almost nonnegative Ricci curvature, Amer. J. Math. 118 (1996), 341-353. Zbl0865.53058MR1385281
37. [LiW] P. Li – J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), 501-534. Zbl1032.58016MR1906784
38. [Loh] N. Lohoué, Estimations asymptotiques des noyaux résolvants du laplacien des formes différentielles sur les espaces symétriques de rang un, de type non compact et applications, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 551-554. Zbl0656.43007MR967359
39. [LR] N. Lohoué – Th. Rychener, Die Resolvente von $\Delta$ auf symmetrischen Räumen vom nichtkompakten Typ, Commun. Math. Helv. 57 (1982), 445-468. Zbl0505.53022MR689073
40. [Lot] J. Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471-510. Zbl0770.58040MR1158345
41. [Maz] R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309-339. Zbl0656.53042MR961517
42. [MM] R. Mazzeo – R. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260-310. Zbl0636.58034MR916753
43. [MP] R. Mazzeo – R. S. Phillips, Hodge theory on hyperbolic manifolds, Duke Math. J. 60 (1990), 509-559. Zbl0712.58006MR1047764
44. [MW] R. Miatello – N. R. Wallach, The resolvent of the Laplacian on locally symmetric spaces, J. Differential Geom. 36 (1992), 663-698. Zbl0766.53044MR1189500
45. [Mok] N. Mok, Harmonic forms with values in locally constant Hilbert bundles, Proceedings of the Conference in Honor of Jean-Piere Kahane (Orsay, 1993), J. Fourier Anal. Appl. special issue (1995), 433-453. Zbl0891.58001MR1364901
46. [MSY] N. Mok – Y. T. Siu – S. K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993), 57-83. Zbl0808.53043MR1223224
47. [Nay] S. Nayatani, Patterson-Sullivan measure and conformally flat metrics, Math. Z. 225 (1997), 115-131. Zbl0868.53024MR1451336
48. [NR] T. Napier – M. Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal. 5 (1995), 809-851. Zbl0860.53045MR1354291
49. [Olb1] M. Olbrich, $L^2$-invariants of locally symmetric spaces, Doc. Math. 7 (2002), 219-237. Zbl1029.58019MR1938121
50. [Olb2] M. Olbrich, Cohomology of convex cocompact groups and invariant distributions on limit sets, preprint, Universität Göttingen. Zbl1006.22012
51. [OT] T. Ohsawa – K. Takegoshi, Hodge spectral sequence on pseudoconvex domains, Math. Z. 197 (1988), 1-12. Zbl0638.32016MR917846
52. [Pan] P. Pansu, Formules de Matsushima, de Garmland et propriété $\left(T\right)$ pour les groupes agissant sur des espaces symétriques ou des immeubles, Bull. Soc. Math. France 126 (1998), 107-139. Zbl0933.22009MR1651383
53. [Pat] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. Zbl0336.30005MR450547
54. [Ped1] E. Pedon, Analyse harmonique des formes différentielles sur l’espace hyperbolique réel, Thèse, Université de Nancy-I, 1997. 
55. [Ped2] E. Pedon, Analyse harmonique des formes différentielles sur l’espace hyperbolique réel I. Transformation de Poisson et fonctions sphériques, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 671-676. Zbl0909.43006MR1641730
56. [Ped3] E. Pedon, Harmonic analysis for differential forms on complex hyperbolic spaces, J. Geom. Phys. 32 (1999), 102-130. Zbl0935.22010MR1724174
57. [Ped4] E. Pedon, The differential form spectrum of quaternionic hyperbolic spaces, preprint, Université de Reims, 2004, to appear in Bull. Sci. Math. Zbl1070.22005MR2126824
58. [Qui1] J.-F. Quint, Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002), 776-809. Zbl1169.22300MR1935549
59. [Qui2] J.-F. Quint, Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv. 77 (2002), 563-608. Zbl1010.22018MR1933790
60. [RS] M. Reed – B. Simon, “Methods of Modern Mathematical Physics”, Vol. IV, Analysis of Operators, Academic Press, 1979. 
61. [Sha] Y. Shalom, Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math. (2) 152 (2000), 113-182. Zbl0970.22011MR1792293
62. [SS] Z. Shen – C. Sormani, The codimension one homology of a complete manifold with nonnegative Ricci curvature, Amer. J. Math. 123 (2001), 515-524. Zbl1024.53027MR1833151
63. [Sul1] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171-202. Zbl0439.30034MR556586
64. [Sul2] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), 327-351. Zbl0615.53029MR882827
65. [Var] N. Varopoulos, Small time Gaussian estimates of heat diffusion kernels I. The semi-group technique, Bull. Sci. Math. 113 (1989), 253-277. Zbl0703.58052MR1016211
66. [VZ] D. A. Vogan – G. J. Zuckerman, Unitary representations with non-zero cohomology, Compositio Math. 53 (1984), 51-90. Zbl0692.22008MR762307
67. [Wal] N. R. Wallach, “Harmonic Analysis on Homogeneous Spaces”, Dekker, 1973. Zbl0265.22022MR498996
68. [Wan1] X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), 671-688. Zbl1053.53030MR1879811
69. [Wan2] X. Wang, On the $L^2$-cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2002), 311-327. Zbl1221.58023MR1944573
70. [Yeg] N. Yeganefar, Sur la $L^2$-cohomologie des variétés à courbure négative, Duke Math. J. 122 (2004), 145-180. Zbl1069.58013MR2046810
71. [Yue] C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc. 348 (1996), 4965-5005. Zbl0864.58047MR1348871

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