On boundary value problems for Einstein metrics

Anderson, Michael T

doi:10.2140/gt.2008.12.2009

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On boundary value problems for Einstein metrics

Michael T Anderson

Geometry & Topology 12 (2008) 2009–2045

DOI: 10.2140/gt.2008.12.2009

Abstract

On any given compact manifold $M^{n + 1}$ with boundary $\partial M$ , 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 $π_{1} (M, \partial M) = 0$ . 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.

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

Volume 12, issue 4 (2008)