Recent Progress on the Geometry of Univalence Crtiteria

and that the Schwarzian is identically zero exactly for Möbius transformations. Let D denote the unit disk. There has been progress in several areas, but the innovations we treat here come primarily from an injectivity criterion for conformal, local diffeomorphisms of an n-dimensional Riemannian manifold into the n-sphere. The criterion involves a generalization of the Schwarzian derivative which depends both on the conformal factor of the mapping and on the underlying Riemannian metric. The scalar curvature of the metric and the metric diameter of the manifold enter as bounds for the Schwarzian. A majority of the known classical univalence criteria follow from this general result. The proof of the general criterion synthesizes several key ingredients that are present in the proofs of many classical criteria, most particularly the Sturm comparison theorem for second order ∗Both authors were supported in part by FONDECYT grant 1971055.