On boundary value problems for Einstein metrics

Abstract

On any given compact manifold $M^{n+1}$ with boundary $\partial M$, it is proved that the moduli space $\mathcal{ℰ}$ of Einstein metrics on $M$, if non-empty, is a smooth, infinite dimensional Banach manifold, at least when ${\pi }_1\left(M,\partial M\right)=0$. Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on $\partial M$ 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.