Cross-ratio distortion and Douady-Earle extension: I. A new upper bound on quasiconformality

Abstract

Let f be an orientation-preserving circle homeomorphism and Φ (f) the Douady–Earle extension of f, and let ||f||cr be the cross-ratio distortion norm of f and K(Φ (f)) be the maximal dilatation of Φ (f). As a consequence of results in [A. Douady and C. J. Earle, ‘Conformally natural extension of homeomorphisms of circle’, Acta Math. 157 (1986) 23–48 and M. Lehtinen, ‘The dilatation of Beurling–Ahlfors extensions of quasisymmetric functions’, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983) 187–191], ln K(Φ (f)) has an upper bound depending on ||f||cr exponentially. In this paper, we first show that ln K(Φ (f)) has an upper bound depending on ||f||cr linearly. Then we extensively study the Douady–Earle extension of a very simple map fλ which depends on a real non-negative parameter λ. For this example, we show that, as λ→∞, (1) our new upper bound on ln K(Φ (fλ)) is substantially smaller than the one given by Douady and Earle in terms of the maximal dilatation of the extremal extension of fλ, and (2) the Douady–Earle extension Φ (fλ) stays exponentially far away from being extremal. Finally, we show that, in general, our upper bound on ln K(Φ (f)) implies the one of Douady and Earle when ||f||cr is large enough.