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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.

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Are you willing to make modifications to the psh functions? e.g. add constants, compose them with convex increasing functions, etc? or do you really want to keep $\psi_1$ and $\psi_2$ as they are? – Sam Lisi May 7 '14 at 15:47
Adding constants or composing with a convex function would be OK as long as we have control on the derivative of this function (at infinity). – Reza Rezazadegan May 7 '14 at 17:26

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