On the differential form spectrum of hyperbolic manifolds

Carron, Gilles; Pedon, Emmanuel

Carron, Gilles ¹ ; Pedon, Emmanuel ²

¹ Laboratoire de Mathématiques Jean Leray (UMR 6629) Université de Nantes 2 rue de la Houssinière B.P. 92208 44322 Nantes Cedex 3, France
² Laboratoire de Mathématiques (UMR 6056) Université de Reims Moulin de la Housse B.P. 1039 51687 Reims Cedex 2, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 705-747.

Résumé

We give a lower bound for the bottom of the $L^{2}$ differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodge-de Rham laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds.

MR Zbl | 3 citations dans Numdam

Classification : 53C35, 22E40, 34L15, 57T15, 58J50

Affiliations des auteurs :

Carron, Gilles ¹ ; Pedon, Emmanuel ²

@article{ASNSP_2004_5_3_4_705_0,
     author = {Carron, Gilles and Pedon, Emmanuel},
     title = {On the differential form spectrum of hyperbolic manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {705--747},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {4},
     year = {2004},
     mrnumber = {2124586},
     zbl = {1170.53309},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2004_5_3_4_705_0/}
}

TY  - JOUR
AU  - Carron, Gilles
AU  - Pedon, Emmanuel
TI  - On the differential form spectrum of hyperbolic manifolds
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2004
SP  - 705
EP  - 747
VL  - 3
IS  - 4
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2004_5_3_4_705_0/
LA  - en
ID  - ASNSP_2004_5_3_4_705_0
ER  -

%0 Journal Article
%A Carron, Gilles
%A Pedon, Emmanuel
%T On the differential form spectrum of hyperbolic manifolds
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 705-747
%V 3
%N 4
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2004_5_3_4_705_0/
%G en
%F ASNSP_2004_5_3_4_705_0

Carron, Gilles; Pedon, Emmanuel. On the differential form spectrum of hyperbolic manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 705-747. http://archive.numdam.org/item/ASNSP_2004_5_3_4_705_0/

Bibliographie
Cité par

[Alb] P. Albuquerque, Patterson-Sullivan theory in higher rank symmetric spaces, Geom. Funct. Anal. 9 (1999), 1-28. | MR | Zbl

[AAR] G. E. Andrews - R. Askey - R. Roy, “Special functions", Encyclopedia Math. Appl. 71., Cambridge Univ. Press, 1999. | MR | Zbl

[ADY] J.-Ph. Anker - E. Damek - C. Yacoub, Spherical analysis on harmonic $A N$ groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 643-679. | Numdam | MR | Zbl

[AJ] J.-Ph. Anker - L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035-1091. | MR | Zbl

[BCG1] G. Besson - G. Courtois - S. Gallot, Lemme de Schwarz réel et applications géométriques, Acta Math. 183 (1999), 145-169. | MR | Zbl

[BCG2] G. Besson - G. Courtois - S. Gallot, Hyperbolic manifolds, almagamated products and critical exponents, C. R. Math. Acad. Sci. Paris 336 (2003), 257-261. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[Bow] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-25. | Numdam | MR | Zbl

[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. | MR | Zbl

[BO1] U. Bunke - M. Olbrich, The spectrum of Kleinian manifolds, J. Funct. Anal. 172 (2000), 76-164. | MR | Zbl

[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. | MR | Zbl

[BO3] U. Bunke - M. Olbrich, Towards the trace formula for convex-cocompact groups, preprint, 2000. | MR

[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. | MR | Zbl

[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. | MR | Zbl

[Car] G. Carron, Une suite exacte en $L^{2}$ -cohomologie, Duke Math. J. 95 (1998), 343-372. | MR | Zbl

[CG] J. Cheeger - D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971-1972), 119-128. | MR | Zbl

[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. | Numdam | MR | Zbl

[CM] A. Connes - H. Moscovici, The $L^{2}$ -index theorem for homogeneous spaces of Lie groups, Ann. of Math. (2) 115 (1982), 292-330. | MR | Zbl

[Cor] K. Corlette, Hausdorff dimensions of limit sets, Invent. Math. 102 (1990), 521-541. | MR | Zbl

[Don] H. Donnelly, The differential form spectrum of hyperbolic space, Manuscripta Math. 33 (1981), 365-385. | MR | Zbl

[DF] H. Donnelly - Ch. Fefferman, $L^{2}$ -cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), 593-618. | MR | Zbl

[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. | MR | Zbl

[EMM] Ch. Epstein - R. Melrose - G. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1-106. | MR | Zbl

[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.

[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. | MR | Zbl

[HV] P. De La Harpe - A. Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 Soc. Math. France, (1989). | Numdam | Zbl

[Heb] E. Hebey, “Sobolev spaces on Riemannian manifolds”, Lecture Notes in Math., 1635, Springer, 1996. | MR | Zbl

[Hel] S. Helgason, “Differential Geometry, Lie groups, and Symmetric Spaces”, Academic Press, 1978. | MR | Zbl

[Ize] H. Izeki, Limit sets of Kleinian groups and conformally flat Riemannian manifolds, Invent. Math. 122 (1995), 603-625. | MR | Zbl

[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. | Numdam | MR | Zbl

[Kna] A. W. Knapp, “Representation Theory of Semisimple Groups. An Overview Based on Examples”, Princeton Math. Ser. 36, Princeton Univ. Press, 1986. | MR | Zbl

[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. | MR | Zbl

[Leu] E. Leuzinger, Critical exponents of discrete groups and $L^{2}$ -spectrum, Proc. Amer. Math. Soc. 132 (2004), 919-927. | MR | Zbl

[Li] P. Li, On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math. 99 (1990), 579-600. | MR | Zbl

[LiR] P. Li - M. Ramachandran, Kähler manifolds with almost nonnegative Ricci curvature, Amer. J. Math. 118 (1996), 341-353. | MR | Zbl

[LiW] P. Li - J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), 501-534. | MR | Zbl

[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. | MR | Zbl

[LR] N. Lohoué - Th. Rychener, Die Resolvente von $Δ$ auf symmetrischen Räumen vom nichtkompakten Typ, Commun. Math. Helv. 57 (1982), 445-468. | MR | Zbl

[Lot] J. Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471-510. | MR | Zbl

[Maz] R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309-339. | MR | Zbl

[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. | MR | Zbl

[MP] R. Mazzeo - R. S. Phillips, Hodge theory on hyperbolic manifolds, Duke Math. J. 60 (1990), 509-559. | MR | Zbl

[MW] R. Miatello - N. R. Wallach, The resolvent of the Laplacian on locally symmetric spaces, J. Differential Geom. 36 (1992), 663-698. | MR | Zbl

[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. | MR | Zbl

[MSY] N. Mok - Y. T. Siu - S. K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993), 57-83. | MR | Zbl

[Nay] S. Nayatani, Patterson-Sullivan measure and conformally flat metrics, Math. Z. 225 (1997), 115-131. | MR | Zbl

[NR] T. Napier - M. Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal. 5 (1995), 809-851. | MR | Zbl

[Olb1] M. Olbrich, $L^{2}$ -invariants of locally symmetric spaces, Doc. Math. 7 (2002), 219-237. | MR | Zbl

[Olb2] M. Olbrich, Cohomology of convex cocompact groups and invariant distributions on limit sets, preprint, Universität Göttingen. | Zbl

[OT] T. Ohsawa - K. Takegoshi, Hodge spectral sequence on pseudoconvex domains, Math. Z. 197 (1988), 1-12. | MR | Zbl

[Pan] P. Pansu, Formules de Matsushima, de Garmland et propriété $(T)$ pour les groupes agissant sur des espaces symétriques ou des immeubles, Bull. Soc. Math. France 126 (1998), 107-139. | Numdam | MR | Zbl

[Pat] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. | MR | Zbl

[Ped1] E. Pedon, Analyse harmonique des formes différentielles sur l'espace hyperbolique réel, Thèse, Université de Nancy-I, 1997.

[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. | MR | Zbl

[Ped3] E. Pedon, Harmonic analysis for differential forms on complex hyperbolic spaces, J. Geom. Phys. 32 (1999), 102-130. | MR | Zbl

[Ped4] E. Pedon, The differential form spectrum of quaternionic hyperbolic spaces, preprint, Université de Reims, 2004, to appear in Bull. Sci. Math. | MR | Zbl

[Qui1] J.-F. Quint, Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002), 776-809. | MR | Zbl

[Qui2] J.-F. Quint, Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv. 77 (2002), 563-608. | MR | Zbl

[RS] M. Reed - B. Simon, “Methods of Modern Mathematical Physics”, Vol. IV, Analysis of Operators, Academic Press, 1979.

[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. | MR | Zbl

[SS] Z. Shen - C. Sormani, The codimension one homology of a complete manifold with nonnegative Ricci curvature, Amer. J. Math. 123 (2001), 515-524. | MR | Zbl

[Sul1] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171-202. | Numdam | MR | Zbl

[Sul2] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), 327-351. | MR | Zbl

[Var] N. Varopoulos, Small time Gaussian estimates of heat diffusion kernels I. The semi-group technique, Bull. Sci. Math. 113 (1989), 253-277. | MR | Zbl

[VZ] D. A. Vogan - G. J. Zuckerman, Unitary representations with non-zero cohomology, Compositio Math. 53 (1984), 51-90. | Numdam | MR | Zbl

[Wal] N. R. Wallach, “Harmonic Analysis on Homogeneous Spaces”, Dekker, 1973. | MR | Zbl

[Wan1] X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), 671-688. | MR | Zbl

[Wan2] X. Wang, On the $L^{2}$ -cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2002), 311-327. | MR

[Yeg] N. Yeganefar, Sur la $L^{2}$ -cohomologie des variétés à courbure négative, Duke Math. J. 122 (2004), 145-180. | MR | Zbl

[Yue] C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc. 348 (1996), 4965-5005. | MR | Zbl