A notion of extremal analytic discs related to interpolation in the ball
J. Globevnik showed that one can find a proper analytic disc through any discrete subset of the unit ball. We want to characterize those discs such that the preimages in the disc of the points in the ball are in a sense as close together as possible. More precisely, we want the preimage sequence to be the least interpolating possible (in the sense of L. Carleson). Since a sequence in the disc is interpolating iff \(\delta\) is positive, where \(\delta\) is the infimum taken over all base points in the sequence of the product of the invariant (Gleason) distances from that point to all the remaining ones, we are looking for an analytic disc which would minimize \(\delta\) for the preimage sequence. In the case of a finite sequence, we show that -- if such an extremal disc exists, it must be quite regular -- there does exist an extremal disc for a large set of finite sequences -- it can fail to exist even in the case of some sequence of three points, which are characterized explicitly.
scientific article; zbMATH DE number 701007