Revistes Catalanes amb Accés Obert (RACO)

Continuity and convergence properties of extremal-interpolating disks

P. J. Thomas


Let $a$ be a sequence of points in the unit ball of $\Bbb C^n$. Eric Amar and the author have introduced the nonnegative quantity $\rho(a)=\inf_{\alpha}\inf_k \prod_{j:j\ne k}d_G(\alpha_j,\alpha_k)$, where $d_G$ is the Gleason distance in the unit disk and the first infimum is taken over all sequences $\alpha$ in the unit disk which map to $a$ by a map from the disk to the ball.

The value of $\rho(a)$ is related to whether $a$ is an interpolating sequence with respect to analytic disks passing through it, and if $a$ is an interpolating sequence in the ball, then $\rho(a) > 0$.

In this work, we show that $\rho (a)$ can be obtained as the limit of the same quantity for the truncated finite sequences, and that $\rho (a)$ depends continuously on $a$ when $a$ is finite. Furthermore, we describe some of the behavior of the minimizing sequences of maps involved in the extremal problem used to define $\rho$.

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