Conformal fitness and uniformization of holomorphically moving disks (Q2796514)

Consider a family of topological disks (subsets of the Riemann sphere homemorphic to disks) \(U_t\) parameterized by a complex number \(t\), together with a holomorphically varying marked point \(c_t\in U_t\). Under the assumption that their boundaries undergo a holomorphic motion, the author characterizes the Riemann maps \(g_t\) from the unit disk to \(U_t\) mapping \(c_t\) to \(0\) which can be chosen to depend holomorphically on \(t\). This characterization includes: extension of the holomorphic motion preserving the harmonic measure, properties of the conformal radius and behaviour of intrinsic rotations.NEWLINENEWLINE For the proofs, properties of conformal maps from the unit disk are used: existence of radial limits almost everywhere and the fact that this limit is almost everywhere one-to-one or at worst two-to-one. Prime ends and their principal sets are also discussed. Precise proofs are given for each result.NEWLINENEWLINEThe article also contains a discussion for the static case (i.e., without parameters). In this case it is assumed that a homeomorphism (not necessarily locally connected) between boundaries of topological disks is given and one asks whether this homeomorphism extends to a conformal map. The notion of extension here is a bit subtle: it is not a continuous extension but only an extension with respect to landing rays.NEWLINENEWLINEAn example is given showing that, interestingly, the characterization in terms of preservation of the harmonic measure fails in the static case, because a set of positive Lebesgue measure of internal angles can be mapped two-to-one by the radial limit map.