On any given compact manifold with boundary , it is proved that the moduli space of Einstein metrics on , if non-empty, is a smooth, infinite dimensional Banach manifold, at least when . Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on are smooth, but not Fredholm. Instead, one has natural mixed boundary-value problems which give Fredholm boundary maps.
These results also hold for manifolds with compact boundary which have a finite number of locally asymptotically flat ends, as well as for the Einstein equations coupled to many other fields.
"On boundary value problems for Einstein metrics." Geom. Topol. 12 (4) 2009 - 2045, 2008. https://doi.org/10.2140/gt.2008.12.2009