Retracts in Polydisk and Analytic Varieties with the H ^∞-Extension Property

Abstract

This article mainly concerns retracts in polydisk, analytic varieties with the H ^∞-extension property and the three-point Pick problem on $\mathbb{D}^{3}$ . Arising in the study of Nevanlinna-Pick interpolation on the bidisk, Agler and McCarthy recently discovered a remarkable theorem which characterizes subsets in the bidisk with the polynomial extension property, and in this case, these subsets are retracts. To study H ^∞-extensions of holomorphic functions from subvarieties of polydisk, one naturally is concerned with retracts in polydisk. Under certain mild assumptions, it is shown that subvarieties with H ^∞-extension property are exactly retracts. Furthermore, we apply our argument to determine those retracts whose retractions are unique. In particular, a retract in $\mathbb{D}^{2}$ having at least two different retractions is exactly a balanced disk. As an application, we give a sufficient condition of the uniqueness of the solution for the three-point Pick problem on $\mathbb{D}^{3}$ .