Geometry of uniqueness varieties for a three-point Pick problem in $\mathbb{D}^3$
Motivated by the recent progress of research on extending holomorphic functions defined on subvarieties of classical domains and its connections to the 3-point Pick interpolation, we study a special class of two-dimensional algebraic subvarieties $M_α$ of the unit tridisc, defined as the sets $$\lbrace (z_1,z_2,z_3)\in \mathbb{D}^3:α_1z_1+α_2z_2+α_3z_3=\overlineα_1z_2z_3+\overlineα_2z_1z_3+\overlineα_3z_1z_2\rbrace.$$ In this paper we show that given non-degenerated extremal maximal $3$-point Pick problem there exists an $α$ such that $M_α$ appears as its uniqueness variety. We also describe several geometric properties of $M_α$ and show the biholomorphic equivalence between any two surfaces $M_α$ and $M_β$, where the triples $α$ and $β$ satisfy the so called triangle inequality.