Imagine $V$ is a complex vector space and $U_1\subset U_2\subset V$ two Euclidean balls. Let $\psi,\psi_1$ be strictly plurisubharmonic functions on $V$ and $U_1$ respectively.
Question: What are the obstructions to the existence of a strictly plurisubharmonic function on $V$ which equals $\psi_1$ on $U_1$ and $\psi$ on the complement of $U_2$?
In the specific situation I have in mind, $\psi$ is a product structure and $\psi_1$ is the pullback of $\psi$ by a biholomorphic map of the ball. Also both functions have a compact critical point set.
Terminology: A function $f$ on a complex manifold is strictly plurisubharmonic if $-dd^c f$ is a sympectic form on the manifold. If in addition, $f$ is proper and bounded blow, we call it a Stein structure.
