#### Volume 12, issue 4 (2008)

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On boundary value problems for Einstein metrics

### Michael T Anderson

Geometry & Topology 12 (2008) 2009–2045
##### Abstract

On any given compact manifold ${M}^{n+1}$ with boundary $\partial M$, it is proved that the moduli space $\mathsc{ℰ}$ 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.

##### Keywords
Einstein metrics, elliptic boundary, value problems
##### Mathematical Subject Classification 2000
Primary: 58J05, 58J32
Secondary: 53C25