Title: Curvature-Decreasing Maps are Volume-Decreasing

Giving a lower bound of the minimal volume of a manifold in terms of the simplicial volume, M. Gromov obtained a generalization of the Gauss-Bonnet-Chern-Weil formulas and conjectured that the minimal volume of a hyperbolic manifold is achieved by the hyperbolic metric. We proved this conjecture via an analogue of the Schwarz's lemma in the non complex case: if the curvature of X is negative and not greater than the one of Y, then any homotopy class of maps from Y to X contains a map which contracts volumes. We give a construction of this map which, under the assumptions of Mostow's rigidity theorems, is an isometry, providing a unified proof of these theorems. It moreover proves that the moduli space of Einstein metrics, on any compact 4-dimensional hyperbolic manifold reduces to a single point.\\ Assuming that X is a compact negatively curved locally symmetric manifold, and without any curvature assumption on Y, another version of the real Schwarz's lemma provides a sharp inequality between the entropies of Y and X . This answers conjectures of A. Katok and M. Gromov. It implies that Y and X have the same dynamics 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.\\

1991 Mathematics Subject Classification:

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