A characterization of domains quasiconformally equivalent to the unit ball (Q1337429)

It is an open question whether two Euclidean domains are quasiconformally equivalent whenever their Royden compactifications are homeomorphic. This paper settles the question affirmatively for the special case in which one of the domains is the unit ball and thus provides a form of the Riemann mapping theorem for higher dimensions. An example is presented which clarifies the structure of the Royden compactification of a typical domain not quasiconformally equivalent to the unit ball.