Let $M$ be a closed complex Kähler manifold, $dim\_{\mathbb C} M = n\geq 2$, with a Kähler form $\omega$. Assume $U\subset M$ is a Stein domain with a smooth boundary and $f: U\to [0;1]$ is a smooth exhausting function for $U$ such that $f^{-1} (1) = \partial U$ and $L:=f^{-1} (0)$ is a smooth totally real closed submanifold of $U$, $dim\_{\mathbb R} L = n$. Does there exist a (smooth) strictly $\omega$-plurisubharmonic function $\phi: M\to {\mathbb R}$ such that $C:=\min\_{\partial U} \phi - \max\_L \phi$ is arbitrarily large?


Note that taking an arbitrary smooth extension of $f$ to $M$ and multiplying it by a sufficiently small positive constant, one can find $\phi$ with a positive, albeit small, $C$.



Restrictions of strictly $\omega$-plurisubharmonic functions to a Stein domain in a closed Kahler manifold
An answer courtesy of Vincent Guedj (for the terminology see the book "Degenerate Complex Monge-Ampère Equations" by V.Guedj and A.Zeriahi):


No, $C$ cannot be made arbitrarily large - there is an upper bound on it depending only on $L$.


The proof goes as follows.


Since $L$ is totally real, it is not locally pluripolar (this is an old result of
Sadullaev from the 1970s, also see [here](https://arxiv.org/pdf/1207.3312.pdf) for a more precise result). Consequently, $L$ is not $PSH (X,\omega)$-polar (indeed, if there had existed a locally defined $\omega$-psh function equal to $-\infty$ locally on $L$, then adding to it a local Kähler potential of $\omega$ one would have got a locally defined psh function equal to $-\infty$ locally on $L$, which is impossible since $L$ is not locally pluripolar).


Now assume that all smooth $\omega$-psh functions $\phi$ from the question are normalized so that $\max\_L \phi = 0$. Then, since $L$ is not $PSH (X,\omega)$-polar, Corollary 9.18 from the book quoted above implies that there exists $K>0$, depending on $L$, such that $\max\_M \phi \leq K$ for all such $\phi$. (Strictly speaking, Corollary 9.18 yields an upper bound on the $L^1$-norm of $\phi$ but using Proposition 8.5 in the book one translates it to an upper bound on $\max\_M \phi$). Clearly, for all such $\phi$ the constant $K$ bounds $C$ from above:
$$C=\min\_{\partial U} \phi − \max\_L \phi = \min\_{\partial U} \phi \leq \max\_M \phi\leq K.$$


