The Schwarzian derivative and quasiconformal reflections on S^n

It is well known that the Schwarzian derivative of an analytic map defined in a domain in the plane is closely related to global univalence and quasiconformal extension. Osgood and Stowe have recently found a generalization of the Schwarzian derivative for conformal local diffeomorphisms between Riemannian manifolds in arbitrary dimension. They establish a univalence criterion for such maps when the target is the sphere Sn. The condition is expressed as an inequality involving the norm of the generalized Schwarzian and quantities that depend on the geometry of the domain manifold. From this result it is possible to recover many injectivity criteria in the unit disc, including two classical conditions of Nehari. In connection with this work, we employ in this paper the techniques developed by C. Epstein to construct quasiconformal reflections in Sn via hypersurfaces in hyperbolic n + 1-space. Our main result shows that a strong form of the univalence criterion of Osgood and Stowe implies the existence of an orientation-reversing quasiconformal diffeomorphism of Sn which fixes pointwise the boundary of the image of the map.