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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.

MSC:
53C20 Global Riemannian geometry, including pinching
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
53C24 Rigidity results
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