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Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds
This paper studies several aspects of asymptotically hyperbolic Einstein
metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for
such metrics and establish uniqueness under natural conditions on the boundary
data. By examination of explicit black hole metrics, it is shown that neither
uniqueness nor finiteness holds in general for AH Einstein metrics with a
prescribed conformal infinity. We then describe natural conditions which are
sufficient to ensure finiteness.Comment: 33pp, gap in one proof fixed, exposition improved. To appear in
Advances in Mat
On the local rigidity of Einstein manifolds with convex boundary
Let (M, g) be a compact Einstein manifold with non-empty boundary. We prove
that Killing fields at the boundary extend to Killing fields of any (M, g)
provided the boundary is weakly convex and a simple condition on the
fundamental group holds. This gives a new proof of the classical infinitesimal
rigidity of convex surfaces in Euclidean space and generalizes the result to
Einstein metrics of any dimension.Comment: withdrawn for reconstruction; error in "stability argument" on p.1
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