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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.

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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