CURVATURES AND ANISOMETRY OF MAPS by

Abstract. — We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The proofs relate to a conjectural isoperimetric inequality for manifolds whose curvature is bounded above, and to a higherdimensional generalization of the Schwarz-Ahlfors lemma. 1