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Summary: BOUNDARY REGULARITY, UNIQUENESS AND NON-UNIQUENESS FOR
AH EINSTEIN METRICS ON 4-MANIFOLDS
MICHAEL T. ANDERSON
Abstract. This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly
on 4-manifolds. We prove boundary regularity (at in nity) for such metrics and establish unique-
ness under natural conditions on the boundary data. By examination of explicit black hole metrics,
it is shown that neither uniqueness nor niteness holds in general for AH Einstein metrics with a
prescribed conformal in nity. We then describe natural conditions which are suÆcient to ensure
niteness.
Contents
0. Introduction. 1
1. Conformally Compact Einstein Metrics. 4
2. Boundary Regularity. 8
3. Uniqueness. 13
4. Non-Uniqueness. 19
5. Cusp Formation and Hyperbolic Manifolds. 24
References 32
0. Introduction.
In this paper, we study several aspects of asymptotically hyperbolic (AH) Einstein metrics on
an open 4-manifold M with compact boundary @M: These metrics are complete Einstein metrics
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