6 Rollback to Revision 4

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 $i$, $\left\|\bar{\partial}f_i\right\| \|\bar{\partial}f_i\| \leq 1$ 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 $\left\|\bar{\partial}f_i\right\| \|\bar{\partial}f_i\| \leq 1$?

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 $i$, $\|\bar{\partial}f_i\| \left\|\bar{\partial}f_i\right\| \leq 1$ 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 $\|\bar{\partial}f_i\| \left\|\bar{\partial}f_i\right\| \leq 1$?

4 Rewrote last line.

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(i.e. ? That is, can we find a locally conformally Kähler, Hermitian-Einstein, etc.)or some other special metric which will give $\|\bar{\partial}f_i\| \leq 1$?

3 Added further conditions on $f_i$.
2 More specific about situation, and removed additional questions which no longer apply.
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