Extension domains.

Abstract

This research is concerned with a problem in higher dimensional quasiconformal mappings and a related problem in the complex plane. Specifically, we study domains D in $\IR\sp{\rm n}$ for which each quasiconformal self map of D extends to a quasiconformal self map of $\IR\sp{\rm n}$. For n = 2, it was shown by Ahlfors and Rickman that a simply connected domain D has the above extension property if and only if D is a quasidisk. Suppose that G is a Jordan domain in $\overline{\IR}\sp2$. We consider the relationship between the geometry of G and the above extension property for D = G $\times$ $\IR$. Vaisala showed that G $\times$ $\IR$ is quasiconformally equivalent B$\sp3$ if and only if G is an inner chordarc domain. For our extension problem we need to consider the geometry of such domains. We show that a Jordan domain $\rm G\subset\IR\sb2$ with locally rectifiable boundary is an inner chordarc domain if and only if for each straight cross-cut $\delta$ = (z, w) of G, $\rm\ell(\gamma)\leq c\vert z{-}w\vert,$ where $\gamma$ is the shorter arc in $\partial$G joining z, w. Next G is an inner chordarc domain if and only if G is a John domain and $\partial$G is regular. Finally, a characterization in terms of the corresponding Riemann map is given. Our extension results are the following. If G is a bounded inner chordarc Jordan domain in $\IR\sp2$, then G $\times$ $\IR$ does not have the extension property for the class of quasiconformal maps fixing $\infty$. On the other hand, if G is an inner chordarc domain and if G* = $\IR\sp2\\$G, then G* $\times$ $\IR$ has the extension property for quasisymmetric maps. Next if D = G $\times$ $\IR$ is quasiconformally equivalent to B$\sp{3}$, then D has the quasiconformal extension property if and only if D is a quasiball. Finally, using the reflection properties of quasiconformal maps we show that if G is an unbounded quasidisk in $\IR\sp2$, then G $\times$ $\IR$ has the quasiconformal extension property in $\overline{\IR}\sp3$. We also construct a bounded quasidisk G such that G $\times$ $\IR$ has the above property.