On boundary value problems for Einstein metrics

Abstract

On any given compact manifold Mn+1 with boundary ∂M, it is proved that the moduli space □ of Einstein metrics on M, if non empty, is a smooth, infinite dimensional Banach manifold, at least when π1(M,∂M) = 0. Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on ∂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. © 2008 Mathematical Sciences Publishers.