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?