Rigidity and L 2 cohomology of hyperbolic manifolds
[Rigidité et cohomologie L 2 des variétés hyperboliques]
Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2307-2331.

La petitesse de l’exposant critique du groupe fondamental d’une variété hyperbolique implique des résultats d’annulation pour certains espaces de cohomologie et de formes harmoniques L 2 . Nous obtenons ici des résultats de rigidité reliés à ces théorèmes d’annulations. Ceci est une généralisation de résultats déjà connus dans le cas convexe co-compact.

When X=Γ n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L 2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.

DOI : 10.5802/aif.2608
Classification : 58J50, 22E40
Keywords: $L^2$ harmonic form, hyperbolic manifold, critical exponent
Mot clés : formes harmoniques $L^2$, variété hyperbolique, exposant critique
Carron, Gilles 1

1 Université de Nantes Laboratoire de mathématiques Jean Leray 2, rue de la Houssinière BP 92208 44322 Nantes cedex 03 (France)
@article{AIF_2010__60_7_2307_0,
     author = {Carron, Gilles},
     title = {Rigidity and $L^2$ cohomology  of hyperbolic manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2307--2331},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     doi = {10.5802/aif.2608},
     zbl = {1236.53040},
     mrnumber = {2848671},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2608/}
}
TY  - JOUR
AU  - Carron, Gilles
TI  - Rigidity and $L^2$ cohomology  of hyperbolic manifolds
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 2307
EP  - 2331
VL  - 60
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2608/
DO  - 10.5802/aif.2608
LA  - en
ID  - AIF_2010__60_7_2307_0
ER  - 
%0 Journal Article
%A Carron, Gilles
%T Rigidity and $L^2$ cohomology  of hyperbolic manifolds
%J Annales de l'Institut Fourier
%D 2010
%P 2307-2331
%V 60
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2608/
%R 10.5802/aif.2608
%G en
%F AIF_2010__60_7_2307_0
Carron, Gilles. Rigidity and $L^2$ cohomology  of hyperbolic manifolds. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2307-2331. doi : 10.5802/aif.2608. http://archive.numdam.org/articles/10.5802/aif.2608/

[1] Anderson, Michael T. L 2 harmonic forms on complete Riemannian manifolds, Geometry and analysis on manifolds (Katata/Kyoto, 1987) (Lecture Notes in Math.), Volume 1339, Springer, Berlin, 1988, pp. 1-19 | DOI | MR | Zbl

[2] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992 | MR | Zbl

[3] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Lemme de Schwarz réel et applications géométriques, Acta Math., Volume 183 (1999) no. 2, pp. 145-169 | DOI | MR | Zbl

[4] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Hyperbolic manifolds, amalgamated products and critical exponents, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 3, pp. 257-261 | DOI | MR | Zbl

[5] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Rigidity of amalgamated products in negative curvature, J. Differential Geom., Volume 79 (2008) no. 3, pp. 335-387 http://projecteuclid.org/getRecord?id=euclid.jdg/1213798182 | MR | Zbl

[6] Bishop, Christopher J.; Jones, Peter W. Hausdorff dimension and Kleinian groups, Acta Math., Volume 179 (1997) no. 1, pp. 1-39 | DOI | MR | Zbl

[7] Bourdon, Marc Sur le birapport au bord des CAT (-1)-espaces, Inst. Hautes Études Sci. Publ. Math. (1996) no. 83, pp. 95-104 | DOI | Numdam | MR | Zbl

[8] Bourguignon, Jean-Pierre The “magic” of Weitzenböck formulas, Variational methods (Paris, 1988) (Progr. Nonlinear Differential Equations Appl.), Volume 4, Birkhäuser Boston, Boston, MA, 1990, pp. 251-271 | MR | Zbl

[9] Bowen, Rufus Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 11-25 | DOI | Numdam | MR | Zbl

[10] Branson, T. Kato constants in Riemannian geometry, Math. Res. Lett., Volume 7 (2000) no. 2-3, pp. 245-261 | MR | Zbl

[11] Calderbank, David M. J.; Gauduchon, Paul; Herzlich, Marc Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal., Volume 173 (2000) no. 1, pp. 214-255 | DOI | MR | Zbl

[12] Carron, Gilles; Pedon, Emmanuel On the differential form spectrum of hyperbolic manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 3 (2004) no. 4, pp. 705-747 | Numdam | MR | Zbl

[13] Izeki, Hiroyasu Limit sets of Kleinian groups and conformally flat Riemannian manifolds, Invent. Math., Volume 122 (1995) no. 3, pp. 603-625 | DOI | MR | Zbl

[14] Izeki, Hiroyasu; Nayatani, Shin Canonical metric on the domain of discontinuity of a Kleinian group, Séminaire de Théorie Spectrale et Géométrie, Vol. 16, Année 1997–1998 (Sémin. Théor. Spectr. Géom.), Volume 16, Univ. Grenoble I, Saint, 1997–1998, pp. 9-32 | Numdam | Zbl

[15] Kapovich, Michael Homological dimension and critical exponent of Kleinian groups, Geom. Funct. Anal., Volume 18 (2009) no. 6, pp. 2017-2054 | DOI | MR | Zbl

[16] Li, Peter; Wang, Jiaping Complete manifolds with positive spectrum, J. Differential Geom., Volume 58 (2001) no. 3, pp. 501-534 http://projecteuclid.org/getRecord?id=euclid.jdg/1090348357 | MR | Zbl

[17] Mazzeo, Rafe The Hodge cohomology of a conformally compact metric, J. Differential Geom., Volume 28 (1988) no. 2, pp. 309-339 http://projecteuclid.org/getRecord?id=euclid.jdg/1214442281 | MR | Zbl

[18] Mazzeo, Rafe; Phillips, Ralph S. Hodge theory on hyperbolic manifolds, Duke Math. J., Volume 60 (1990) no. 2, pp. 509-559 | DOI | MR | Zbl

[19] Patterson, S. J. The limit set of a Fuchsian group, Acta Math., Volume 136 (1976) no. 3-4, pp. 241-273 | DOI | MR | Zbl

[20] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer-Verlag, New York, 1994 | MR | Zbl

[21] Shalom, Yehuda Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math. (2), Volume 152 (2000) no. 1, pp. 113-182 | DOI | MR | Zbl

[22] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 171-202 | DOI | Numdam | MR | Zbl

[23] Sullivan, Dennis Related aspects of positivity in Riemannian geometry, J. Differential Geom., Volume 25 (1987) no. 3, pp. 327-351 http://projecteuclid.org/getRecord?id=euclid.jdg/1214440979 | MR | Zbl

[24] Wang, Xiaodong On conformally compact Einstein manifolds, Math. Res. Lett., Volume 8 (2001) no. 5-6, pp. 671-688 | MR | Zbl

[25] Wang, Xiaodong On the L 2 -cohomology of a convex cocompact hyperbolic manifold, Duke Math. J., Volume 115 (2002) no. 2, pp. 311-327 | DOI | MR

[26] Yeganefar, Nader Sur la L 2 -cohomologie des variétés à courbure négative, Duke Math. J., Volume 122 (2004) no. 1, pp. 145-180 | DOI | MR | Zbl

[27] Yue, Chengbo Dimension and rigidity of quasi-Fuchsian representations, Ann. of Math. (2), Volume 143 (1996) no. 2, pp. 331-355 | DOI | MR | Zbl

Cité par Sources :