Rendiconti di Matematica e delle Sue Applicazioni (Jan 1998)
A real Schwarz lemma and some applications
Abstract
The “Minimal Volume” of a differentiable manifold was introduced by M. Gromov in order to generalize (in any dimension) the inequalities deduced from the Gauss-Bonnet-Chern-Weil theory of characteristic classes. By bounding the minimal volume in terms of the simplicial volume, M. Gromov gave such a generalization. By computing the simplicial volumes of the hyperbolic manifolds, M. Gromov (with W. Thurston) made this inequality explicit (and revisited Mostow’s rigidity theorem). He conjectured that this inequality might be sharpen, (i.e. that the minimal volume is attained for the hyperbolic metric). We proved this conjecture by settling a real analogue of the Schwarz’s lemma: if X, Y are two manifolds such that the curvature of X is negative and smaller than the one of Y , then any homotopy class of maps from Y to X contains a map which contracts volumes. We give an explicit construction of this application which, under the assumptions of Mostow’s rigidity theorems, occurs to be an isometry, providing a unified proof of these theorems. It moreover proves that the set of all Einstein metrics, on any compact 4-dimensional hyperbolic manifold, reduces to a single point. A modified version of the real Schwarz’s lemma gives a sharp inequality between the entropies of Y and X (provided that X is locally symmetric and that there exists an application of non trivial degree from Y to X). This answers conjectures of M. Gromov and A. Katok about the minimal entropy. As this inequality is an equality iff Y is isometric to X, it implies that Y and X have conjugate geodesic flows iff they are isometric. This also ends the proof of the Lichnerowicz’s conjecture: any negatively curved compact locally harmonic manifold is a quotient of a (noncompact) rank-one-symmetric space.
