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What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have non-complete Kähler-Einstein metrics being open subsets of ${\bf C}P^2$.

One could also ask the much simpler question if there are any more or less explicitly constructed complete metrics on some exotic ${\bf R}^4$s, maybe with bounded curvature or satisfying some other decent condition?

There are some obvious obstructions to good metrics on exotic ${\bf R}^4$s arising from Hopf-Rinow, Cheeger-Gromoll splitting, etc. What is known about "non-obvious" obstructions?

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It is theorem of Robert E Greene that every smooth manifold admits complete riemannian metrics with bounded geometry i.e the curvature and all its covariant derivatives are bounded and the injectivity radius is bounded below by a positive constant . – Mohan Ramachandran Feb 24 '12 at 22:37
Maybe it is worth noting that every (paracompact) manifold of dimension $n \geq 3$ admits a complete metric of scalar curvature -1. This was proved by Aubin (in the compact case) and Bland-Kalka, and reproved (among other things) by Lohkamp in – 680 Mar 2 '12 at 15:16

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