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The real Schwarz Lemma and geometric applications. (Lemme de Schwarz réel et applications géométriques.) (French) Zbl 1035.53038
In the first result of the present article the authors give a Riemannian analogue of the classical Schwarz lemma (that holomorphic maps between complex manifolds with suitable curvature conditions decrease volume): Let \(X\), \(Y\) be complete Riemannian manifolds of the same dimension \(n \geq 3\), such that \(\text{Ricci}_{g_{Y}} \geq -(n-1)g_{Y}\) and \(K_{g_{X}} \leq -1\). Then in any homotopy class of maps \(f:Y \rightarrow X\) and for all \(\varepsilon > 0\), there exists a map \(F_{\varepsilon}\) such that \(| \text{Jac}(F_{\varepsilon})| \leq 1 + \varepsilon\) pointwise. If, moreover, \(X\) and \(Y\) are compact and homotopically equivalent with \(K_{g_{Y}} \leq -1\), there exists \(F_{0}\) of class \(C^{\infty}\) such that \(| \text{Jac}(F_0)| \leq 1\) pointwise; and if \(| \text{Jac}F_0(y))| = 1\) for some \(y\) then \(d_yF_{0}\) is an isometry. The proof is a well-readable version of proof ideas already used by the authors in their articles [Geom. Funct. Anal. 5, 731–799 (1995; Zbl 0851.53032) and Ergodic Theory Dyn. Syst. 16, 623–649 (1996; Zbl 0887.58030)].
The key observation of the present paper is that with the same proof ideas one can obtain a much more general result: In fact, the authors get such maps \(F_{\varepsilon}\) and \(F_{0}\) also if \(X\) and \(Y\) are not necessarily of the same dimension, and these maps depend only on a representation of the fundamental group of \(Y\) in the one of \(X\); more generally, the authors show that one can start with a representation \(\rho : \Gamma \rightarrow \Gamma'\) of discrete isomorphism groups \(\Gamma\) of \(Y\) and \(\Gamma'\) of \(X\) and gets \(\rho\)-equivariant maps \(F_{\varepsilon}\) resp. \(F_{0}\) still satisfying inequalities of the type \(| \text{Jac}F_{\varepsilon}| \leq (\frac{\delta (\Gamma)}{p-1}(1+\varepsilon))^{p}\), where \(p=\text{dim}(Y)\) and \(\delta (\Gamma)\) is the critical exponent of the Poincaré series of \(\Gamma\).
As a corollary the authors give a rigidity result for quasi-Fuchsian representations for a cocompact lattice \(\Gamma \subset \text{Isom}(Y)\) of a complex hyperbolic space \(Y\), which yields that there exists a constant \(C>0\) such that any such representation \(\rho\) satisfying \(\delta (\rho (\Gamma))< (1+C)\delta (\rho_{0}(\Gamma))\) with a totally geodesic representation \(\rho_{0}\) is itself totally geodesic. Using a similar idea they also give a very nice (and purely differential geometric) proof of an \(L^{2}\)-version of a conjecture of Gromov, which sais that there exists a universal constant \(C(m,d)\) such that every compact complex subvariety of dimension \(d\) of a quotient of a complex hyperbolic space of dimension \(m\), such that the \(L^{2}\)- and the \(L^{2d}\)-norm of the second fundamental form is bounded by \(C(m,d)\), is already totally geodesic.

53C20 Global Riemannian geometry, including pinching
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
53C24 Rigidity results
Full Text: DOI
[1] [BCG1]Besson, G., Courtois, G. &Gallot, S., Entropies et rigidités des espaces localement symétriques de courbure strictement négative.Geom. Funct. Anal., 5 (1995), 731–799. · Zbl 0851.53032 · doi:10.1007/BF01897050
[2] [BCG2] –, Minimal entropy and Mostow’s rigidity theorems.Ergodic Theory Dynam. Systems, 16 (1996), 623–649. · Zbl 0887.58030 · doi:10.1017/S0143385700009019
[3] [Be]Besse, A. L.,Einstein Manifolds. Ergeb. Math. Grenzgeb. (3), 10. Springer-Verlag, Berlin-New York, 1987.
[4] [Bou]Bourdon, M., Sur le birapport au bord des CAT()-espaces.Inst. Hautes Études Sci. Publ. Math., 83 (1996), 95–104. · Zbl 0883.53047 · doi:10.1007/BF02698645
[5] [Bow]Bowen, R., Haussdorff dimension of quasi-circles.Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11–25. · Zbl 0439.30032 · doi:10.1007/BF02684767
[6] [BP]Benedetti, R. &Petronio, C.,Lectures on Hyperbolic Geometry. Universitext. Springer-Verlag, Berlin, 1992.
[7] [BZ]Burago, Yu. D. &Zalgaller, V. A.,Geometric Inequalities. Grundlehren Math. Wiss., 285. Springer-Verlag, Berlin-New York, 1988.
[8] [Co]Corlette, K., Archimedean superrigidity and hyperbolic geometry.Ann. of Math., 135 (1992), 165–182. · Zbl 0768.53025 · doi:10.2307/2946567
[9] [CT]Carlson, J. &Toledo, D., Harmonic mappings of Kähler manifolds to locally symmetric spaces.Inst. Hautes Études Sci. Publ. Math., 69 (1989), 173–201. · Zbl 0695.58010 · doi:10.1007/BF02698844
[10] [DE]Douady, E. &Earle, C., Conformally natural extension of homeomorphisms of the circle.Acta Math., 157 (1986), 23–48. · Zbl 0615.30005 · doi:10.1007/BF02392590
[11] [ES]Eells, J. &Sampson, J. H., Harmonic mappings of Riemannian manifolds.Amer. J. Math., 86 (1964), 109–160. · Zbl 0122.40102 · doi:10.2307/2373037
[12] [GH]Ghys, E. &Harpe, P. de la (éditeurs),Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988). Progr. Math., 83. Birkhäuser Boston, Boston, MA, 1990. · Zbl 0731.20025
[13] [GM]Goldman, W. &Millson, J., Local rigidity of discrete groups acting on complex hyperbolic space.Invent. Math., 88 (1987), 495–520. · Zbl 0627.22012 · doi:10.1007/BF01391829
[14] [Gr1]Gromov, M., Volume and bounded cohomology.Inst. Hautes Études Sci. Publ. Math., 56 (1981), 213–307.
[15] [Gr2]–, Asymptotic invariants of infinite groups, inGeometric Group Theory, Vol. 2 (Sussex, 1991), pp. 1–295. London Math. Soc. Lecture Note Ser., 182. Cambridge Univ. Press, Cambridge, 1993.
[16] [HS]Hoffmann, W. &Spruck, J., Sobolev and isoperimetric inequalities for Riemannian submanifolds.Comm. Pure Appl. Math., 27 (1975), 715–727; Erratum.Comm. Pure Appl. Math., 27 (1975), 765–766. · Zbl 0295.53025 · doi:10.1002/cpa.3160270601
[17] [JY]Jost, J. &Yau, S. T., Harmonic mappings and Kähler manifolds.Math. Ann., 262 (1983), 145–166. · Zbl 0527.53041 · doi:10.1007/BF01455308
[18] [Mo]Mok, N.,Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds. Ser. Pure Math., 6. World Sci. Publishing, Singapore, 1989. · Zbl 0912.32026
[19] [O]Oliveira Filho, G. de, Compactification of minimal submanifolds of hyperbolic space.Comm. Anal. Geom., 1 (1993), 1–29. · Zbl 0794.53038
[20] [P]Pansu, P., Dimension conforme et sphère à l’infini des variétés à courbure négative.Ann. Acad. Sci. Fenn. Ser. A I Math., 14 (1989), 177–212. · Zbl 0722.53028
[21] [Sa]Sambusetti, A., Best constants for a real Schwarz lemma. Prépublication de l’Institut Fourier no 422. Grenoble, 1998.
[22] [Sh]Shalom, Y., Rigidity and cohomology of unitary representations.Internat. Math. Res. Notices, 16 (1998), 829–849. · Zbl 0911.22009 · doi:10.1155/S1073792898000518
[23] [Si]Siu, Y. T., Complex analyticity of harmonic maps and strong rigidity of complex Kähler manifolds.Ann. of Math., 112 (1980), 73–111. · Zbl 0517.53058 · doi:10.2307/1971321
[24] [SP]Séminaire Palaiseau 1978, Première classe de Chern et courbure de Ricci: preuve de la conjecture de Calabi.Astérisque no 58.
[25] [Su]Sullivan, D., The density at infinity of a discrete group of hyperbolic motions.Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171–202. · Zbl 0439.30034 · doi:10.1007/BF02684773
[26] [Y1]Yue, C. B., Dimension and rigidity of quasi-Fuchsian representations.Ann. of Math., 143 (1996), 331–355. · Zbl 0843.22019 · doi:10.2307/2118646
[27] [Y2]–, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature.Trans. Amer. Math. Soc., 348 (1996), 4965–5005. · Zbl 0864.58047 · doi:10.1090/S0002-9947-96-01614-5
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