Curvatures and anisometry of maps

TL;DR

This paper establishes inequalities quantifying the deviation of local maps between manifolds of differing curvatures from being isometries, focusing on volume-preserving, conformal, and quasi-conformal maps, and relates these to isoperimetric inequalities and a generalized Schwarz-Ahlfors lemma.

Contribution

It introduces new inequalities that measure how non-isometric local maps are between high and low curvature manifolds, extending classical results to broader contexts.

Findings

Derived inequalities for volume-preserving, conformal, and quasi-conformal maps.

Connected these inequalities to isoperimetric conjectures for curved manifolds.

Linked results to a higher-dimensional Schwarz-Ahlfors lemma.

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 higher-dimensional generalization of the Schwarz-Ahlfors lemma.