Curvature-decreasing maps are volume-decreasing. On joint work with G. Besson and G. Courtois (Q1126781)

This article contains a review of solutions of some conjectures obtained by G. Besson, G. Courtois and the author of the present paper. Namely, results concerning the Gromov conjecture (the minimal volume of a hyperbolic manifold is achived by the hyperbolic metric), the conjectures of A. Katok and M. Gromov about the minimal entropy and the conjecture of A. Lichnerowicz (any negatively curved compact locally harmonic manifold is a quotient of a noncompact rank one symmetric space) are considered.