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


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