Finding complete conformal metrics to extend conformal mappings

Abstract. This paper shows how new dierential geometric ap-proaches to univalence criteria involving the Schwarzian derivative can be applied to a classical, but very general, criterion of Nehari. We show how positive solutions to the second order ODE associ-ated to the Schwarzian can be used to construct complete conformal metrics. These lead to explicit formulas for homeomorphic and qua-siconformal extensions of conformal mappings as generalizations of the Ahlfors-Weill extension. 1. Introduction. Let f be analytic in the unit disk D, and let Sf = (f 00=f 0)0 − (1=2)(f 00=f 0)2 be its Schwarzian derivative. Starting with the funda-mental work of Nehari, [11], and ever since the paper of Ahlfors and Weill, [2], univalence criteria involving the Schwarzian derivative have gone hand-in-hand with the phenomenon of quasiconformal extension to the sphere. The neates