Facial behaviour of analytic functions on the bidisk

We prove that if $ϕ$ is an analytic function bounded by 1 on the bidisk and $τ$ is a point in a face of the bidisk at which $ϕ$ satisfies Caratheodory's condition then both $ϕ$ and the angular gradient $\nablaϕ$ exist and are constant on the face. Moreover, the class of all $ϕ$ with prescribed $ϕ(τ)$ and $\nablaϕ(τ)$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.

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