Recent Progress on the Geometry of Univalence Crtiteria M. Chuaqui P. Universidad Cat´olica de Chile B. Osgood ∗ Stanford University 1 Introduction This paper is a survey of some new techniques and new results on sufficient conditions in terms of the Schwarzian derivative for analytic functions defined in the unit disk to be univalent. Along with univalence we consider the questions of quasiconformal and homeomorphic extensions of the mapping. Let f be analytic and locally univalent. Its Schwarzian derivative is Sf =  f ′′ f ′  ′ − 1 2  f ′′ f ′  2 . If u =(f ′ ) −1/2 then u ′′ + 1 2 (Sf )u =0. Conversely if u is a solution of u ′′ + pu =0 (1.1) and f (z )=  z z 0 u −2 (ζ ) dζ (1.2) then Sf =2p. We recall the chain rule S (f ◦ g) = ((Sf ) ◦ g))(g ′ ) 2 + Sg (1.3) and that the Schwarzian is identically zero exactly for M¨obius 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. 1