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**On boundary value problems for Einstein metrics.**
*(English)*
Zbl 1146.58011

Summary: 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 \(\pi_1(M,\partial M) = 0\). 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.

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.

### MSC:

58J05 | Elliptic equations on manifolds, general theory |

58J32 | Boundary value problems on manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

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\textit{M. T. Anderson}, Geom. Topol. 12, No. 4, 2009--2045 (2008; Zbl 1146.58011)

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