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EINSTEIN METRICS WITH PRESCRIBED CONFORMAL INFINITY ON 4-MANIFOLDS
 

Summary: EINSTEIN METRICS WITH PRESCRIBED CONFORMAL INFINITY ON
4-MANIFOLDS
MICHAEL T. ANDERSON
Abstract. This paper considers the existence of conformally compact Einstein metrics on 4-
manifolds. A reasonably complete understanding is obtained for the existence of such metrics
with prescribed conformal in nity, when the conformal in nity is of positive scalar curvature. We
nd in particular that general solvability depends on the topology of the lling manifold. The
obstruction to extending these results to arbitrary boundary values is also identi ed. While most of
the paper concerns dimension 4, some general results on the structure of the space of such metrics
hold in all dimensions.
1. Introduction.
This paper is concerned with the existence of conformally compact Einstein metrics on a given
manifold M with boundary @M . The main results are restricted to dimension 4, although some of
the results hold in all dimensions.
This existence problem was raised by Fe erman and Graham in [19] in connection with a study
of conformal invariants of Riemannian manifolds. More recently, the study of such metrics has
become of strong interest through the AdS/CFT correspondence, relating gravitational theories on
M with conformal eld theories on @M , cf. [18], [38] and references therein.
Let M be a compact, oriented manifold with non-empty boundary @M ; M is assumed to be
connected, but apriori @M may be connected or disconnected. A de ning function  for @M in M

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics