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.