Hi. I have been struggling with this question for a while now.

Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $f: \Omega \longrightarrow \Omega$ such that $f_{|\Omega^\prime}$ is a (real) constant, under which assumptions on $\Omega$ and/or $\Omega^\prime$ can one conclude that $f$ is constant on the whole domain $\Omega$?

The case $n = 1$ is quite classical. For $n>1$, assuming $\Omega$ is a pseudoconvex domain does not really help, and I'm not sure of what can be said about real domains of the zeros of an analytical function of several complex variables.

Thanks for any clue about this.