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Summary: TOPICS IN CONFORMALLY COMPACT EINSTEIN METRICS
MICHAEL T. ANDERSON
1. Introduction.
Conformal compacti cations of Einstein metrics were introduced by Penrose [38], as a
means to study the behavior of gravitational elds at in nity, i.e. the asymptotic behavior of
solutions to the vacuum Einstein equations at null in nity. This has remained a very active
area of research, cf. [27], [19] for recent surveys. In the context of Riemannian metrics, the
modern study of conformally compact Einstein metrics began with the work of Fe erman-
Graham [26], in connection with their study of conformal invariants of Riemannian metrics.
Recent mathematical work in this area has been signi cantly in
uenced by the AdS/CFT
(or gravity-gauge) correspondence in string theory, introduced by Maldacena [36]. We will
only comment brie
y here on aspects of the AdS/CFT correspondence, and refer to [2],
[42], [7] for general surveys.
In this paper, we discuss recent mathematical progress in this area, focusing mainly on
global aspects of conformally compact Einstein metrics and the global existence question
for the Dirichlet problem. One reason for this is that it now appears that the beginnings of
a general existence theory for such metrics may be emerging, at least in dimension 4. Of
course to date there is no general theory for the existence of complete Einstein metrics on
manifolds, with two notable exceptions; the existence theory for Kahler-Einstein metrics
due to Calabi, Yau, Aubin and others, and the existence theory in dimension 3, due to
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