Geometry & Topology 12 (2008) 2009–2045

DOI: 10.2140/gt.2008.12.2009

Abstract

On any given compact manifold Mⁿ⁺¹ with boundary ∂M, it is proved that the moduli space E 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.

Keywords

Einstein metrics, elliptic boundary, value problems

Mathematical Subject Classification

Primary: 58J05, 58J32

Secondary: 53C25

References

Publication

Received: 11 March 2008
Revised: 6 May 2008
Accepted: 9 June 2008
Published: 23 July 2008
Proposed: Tobias Colding
Seconded: Simon Donaldson, David Gabai

Authors