Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). For each $i \in \mathbb{N}$, let $f_i$ be a smooth function $X \to [0, 1]$ such that for every compact set $K \subset X$, there exists $N \in \mathbb{N}$ such that $f_i|_K = 1$ for all $i \geq N$.

Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\|\bar{\partial}f_i\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?

If the answer is no, can we do any better than a generic hermitian metric? That is, can we find a locally conformally Kähler, Hermitian-Einstein, or some other special metric which will give $\|\bar{\partial}f_i\| \leq 1$?

locallyunbounded. – BS. Oct 6 '12 at 8:56