*Rounge's Theorem* states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on $\mathbb{C}P^1$). 


Donaldson wrote a small paper *"Approximation of Instantons"* based off of Taubes' gluing construction for instantons on $S^4$, which gives an analog to Runge's approximation theorem for meromorphic functions. There the statement is that given an open subset $U\subset S^4$ and an instanton $A\_U$ over some principal bundle $P\_U\to U$, there is a sequence of bundles $P\_n\to S^4$ and instantons $A\_n$ (and bundle maps $P\_U\to P\_n$ restricted to $U$) such that the pullbacks of $A\_n$ converge to $A\_U$ (over a slightly smaller subset of $U$).


Now Taubes has a similar construction to build anti-self-dual metrics: Given a compact oriented 4-manifold $X$, there exists ASD metrics on $X\#\_n\mathbb{C}P^2$ (meaning connect sum with $n$ copies of $\mathbb{C}P^2$) for $n>>0$. This motivates:


**In a similar spirit, is there likely to be a (Runge-type) approximation theorem in the realm of ASD metrics on 4-manifolds?** I want to propose:  

Let $X$ be a compact oriented 4-manifold. Given an open subset $U\subset X$ and an ASD metric $g$ on $U$, there is an integer $R$ and a sequence of ASD metrics $g\_n$ on $X\_n=X\#\_{R+n}\mathbb{C}P^2$ and inclusions $U\to X\_n$ such that the pullbacks of $g\_n$ converge to $g$.


My immediate concerns/thoughts:  

1) For this statement to even make sense I wouldn't be able to connect sum $\mathbb{C}P^2$ into the region $U$.  

2) If Taubes' construction can build an ASD metric on $X\_n$ which doesn't touch $g$ on $U$, then my desired statement is true.



Approximation theorem for Anti-Self-Dual Metrics
I think that your point 2) is more or less correct. In his proof Taubes first constructs approximately ASD metrics $g\_N$ on $X \#\_N CP^2$, starting with a metric $g$ on $X$, and this step would not touch the metric on $U$ if $g$ is already ASD there. These will be metrics with 
$$ \lim\_{N\to \infty} \Vert W^+(g\_N)\Vert = 0 $$
in a suitable norm, and $g\_N = g$ on $U$, i.e. no connect sum is performed in $U$. An extra layer of connect sums is performed to ensure invertibility of the linearized problem (as opposed to just decreasing $W^+$), but I suspect that one can do this without touching $U$ as well. 


Then the perturbation to a genuine ASD metric is global, and once $N$ is large enough, you should end up with ASD metrics $g\_{N,ASD}$ on $X\#\_N CP^2$, such that
$$ \lim\_{N\to \infty} \Vert g\_{N,ASD} - g\_N \Vert =0 $$
in a suitable norm. The choice of this norm is such that $g\_{N,ASD}$ will in particular be very close to $g\_N = g$ over $U$ in the $C^0$ sense once $N$ is large. Since both $g\_{N,ASD}$ and $g$ are ASD metrics over $U$, one should be able to get improved regularity there and conclude that on any compact subset $K\subset U$ the metrics $g\_{N,ASD}$ converge to $g$ in any $C^k$ norm.


So in sum I believe that the answer to your question is yes, but of course one would need to go through Taubes' argument to make sure. 


