Volume 12, issue 4 (2008)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On boundary value problems for Einstein metrics

Michael T Anderson

Geometry & Topology 12 (2008) 2009–2045
Abstract

On any given compact manifold Mn+1 with boundary M, it is proved that the moduli space 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 2000
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
Michael T Anderson
Dept of Mathematics
SUNY at Stony Brook
Stony Brook
NY 11794-3651
USA