Rigidity and L 2 cohomology of hyperbolic manifolds
Annales de l'Institut Fourier, Volume 60 (2010) no. 7, p. 2307-2331

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.

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.

DOI : https://doi.org/10.5802/aif.2608
Classification:  58J50,  22E40
Keywords: L 2 harmonic form, hyperbolic manifold, critical exponent
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     pages = {2307-2331},
     doi = {10.5802/aif.2608},
     mrnumber = {2848671},
     zbl = {1236.53040},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_7_2307_0}
}
Rigidity and $L^2$ cohomology  of hyperbolic manifolds. Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2307-2331. doi : 10.5802/aif.2608. http://www.numdam.org/item/AIF_2010__60_7_2307_0/

[1] Anderson, Michael T. L 2 harmonic forms on complete Riemannian manifolds, Geometry and analysis on manifolds (Katata/Kyoto, 1987), Springer, Berlin (Lecture Notes in Math.) Tome 1339 (1988), pp. 1-19 | Article | MR 961469 | Zbl 0652.53030

[2] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Springer-Verlag, Berlin, Universitext (1992) | MR 1219310 | Zbl 0768.51018

[3] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Lemme de Schwarz réel et applications géométriques, Acta Math., Tome 183 (1999) no. 2, pp. 145-169 | Article | MR 1738042 | Zbl 1035.53038

[4] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Hyperbolic manifolds, amalgamated products and critical exponents, C. R. Math. Acad. Sci. Paris, Tome 336 (2003) no. 3, pp. 257-261 | Article | MR 1968269 | Zbl 1026.57013

[5] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Rigidity of amalgamated products in negative curvature, J. Differential Geom., Tome 79 (2008) no. 3, pp. 335-387 http://projecteuclid.org/getRecord?id=euclid.jdg/1213798182 | MR 2433927 | Zbl 1206.53038

[6] Bishop, Christopher J.; Jones, Peter W. Hausdorff dimension and Kleinian groups, Acta Math., Tome 179 (1997) no. 1, pp. 1-39 | Article | MR 1484767 | Zbl 0921.30032

[7] Bourdon, Marc Sur le birapport au bord des CAT (-1)-espaces, Inst. Hautes Études Sci. Publ. Math. (1996) no. 83, pp. 95-104 | Article | Numdam | MR 1423021 | Zbl 0883.53047

[8] Bourguignon, Jean-Pierre The “magic” of Weitzenböck formulas, Variational methods (Paris, 1988), Birkhäuser Boston, Boston, MA (Progr. Nonlinear Differential Equations Appl.) Tome 4 (1990), pp. 251-271 | MR 1205158 | Zbl 0774.35003

[9] Bowen, Rufus Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 11-25 | Article | Numdam | MR 556580 | Zbl 0439.30032

[10] Branson, T. Kato constants in Riemannian geometry, Math. Res. Lett., Tome 7 (2000) no. 2-3, pp. 245-261 | MR 1764320 | Zbl 1039.53033

[11] Calderbank, David M. J.; Gauduchon, Paul; Herzlich, Marc Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal., Tome 173 (2000) no. 1, pp. 214-255 | Article | MR 1760284 | Zbl 0960.58010

[12] Carron, Gilles; Pedon, Emmanuel On the differential form spectrum of hyperbolic manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Tome 3 (2004) no. 4, pp. 705-747 | Numdam | MR 2124586 | Zbl 1170.53309

[13] Izeki, Hiroyasu Limit sets of Kleinian groups and conformally flat Riemannian manifolds, Invent. Math., Tome 122 (1995) no. 3, pp. 603-625 | Article | MR 1359605 | Zbl 0854.53035

[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, Univ. Grenoble I, Saint (Sémin. Théor. Spectr. Géom.) Tome 16 (1997–1998), pp. 9-32 | Numdam | Zbl 0979.53036

[15] Kapovich, Michael Homological dimension and critical exponent of Kleinian groups, Geom. Funct. Anal., Tome 18 (2009) no. 6, pp. 2017-2054 | Article | MR 2491697 | Zbl 1178.30056

[16] Li, Peter; Wang, Jiaping Complete manifolds with positive spectrum, J. Differential Geom., Tome 58 (2001) no. 3, pp. 501-534 http://projecteuclid.org/getRecord?id=euclid.jdg/1090348357 | MR 1906784 | Zbl 1032.58016

[17] Mazzeo, Rafe The Hodge cohomology of a conformally compact metric, J. Differential Geom., Tome 28 (1988) no. 2, pp. 309-339 http://projecteuclid.org/getRecord?id=euclid.jdg/1214442281 | MR 961517 | Zbl 0656.53042

[18] Mazzeo, Rafe; Phillips, Ralph S. Hodge theory on hyperbolic manifolds, Duke Math. J., Tome 60 (1990) no. 2, pp. 509-559 | Article | MR 1047764 | Zbl 0712.58006

[19] Patterson, S. J. The limit set of a Fuchsian group, Acta Math., Tome 136 (1976) no. 3-4, pp. 241-273 | Article | MR 450547 | Zbl 0336.30005

[20] Ratcliffe, John G. Foundations of hyperbolic manifolds, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 149 (1994) | MR 1299730 | Zbl 0809.51001

[21] Shalom, Yehuda Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math. (2), Tome 152 (2000) no. 1, pp. 113-182 | Article | MR 1792293 | Zbl 0970.22011

[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 | Article | Numdam | MR 556586 | Zbl 0439.30034

[23] Sullivan, Dennis Related aspects of positivity in Riemannian geometry, J. Differential Geom., Tome 25 (1987) no. 3, pp. 327-351 http://projecteuclid.org/getRecord?id=euclid.jdg/1214440979 | MR 882827 | Zbl 0615.53029

[24] Wang, Xiaodong On conformally compact Einstein manifolds, Math. Res. Lett., Tome 8 (2001) no. 5-6, pp. 671-688 | MR 1879811 | Zbl 1053.53030

[25] Wang, Xiaodong On the L 2 -cohomology of a convex cocompact hyperbolic manifold, Duke Math. J., Tome 115 (2002) no. 2, pp. 311-327 | Article | MR 1944573 | Zbl pre01941444

[26] Yeganefar, Nader Sur la L 2 -cohomologie des variétés à courbure négative, Duke Math. J., Tome 122 (2004) no. 1, pp. 145-180 | Article | MR 2046810 | Zbl 1069.58013

[27] Yue, Chengbo Dimension and rigidity of quasi-Fuchsian representations, Ann. of Math. (2), Tome 143 (1996) no. 2, pp. 331-355 | Article | MR 1381989 | Zbl 0843.22019