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.


58J05 Elliptic equations on manifolds, general theory
58J32 Boundary value problems on manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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