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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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##### References:
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