Curvatures and anisometry of maps

Benoît Kloeckner (1)

1 IF - Institut Fourier

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 higher-dimensional generalization of the Schwarz-Ahlfors lemma.

Mots clés: volume-preserving maps; conformal maps; isoperimetric inequality; distortion; quasiconformal maps; curvature bounds; sectional curvature; Ricci curvature; Riemannian manifold; anisometry

Domaines: Géométrie différentielle [math.DG]

Benoît Kloeckner. Curvatures and anisometry of maps. Communications in Analysis and Geometry, 2015, 23 (2), pp. 319-348. ⟨10.4310/CAG.2015.v23.n2.a4⟩. ⟨hal-00959852⟩