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ON THE STRUCTURE OF CONFORMALLY COMPACT EINSTEIN METRICS MICHAEL T. ANDERSON
 

Summary: ON THE STRUCTURE OF CONFORMALLY COMPACT EINSTEIN METRICS
MICHAEL T. ANDERSON
Abstract. Let M be an (n+1)-dimensional manifold with non-empty boundary, satisfying 1(M, M) =
0. The main result of this paper is that the space of conformally compact Einstein metrics on M
is a smooth, infinite dimensional Banach manifold, provided it is non-empty. We also prove full
boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem
for such metrics with prescribed metric and stress-energy tensor at conformal infinity, again in
dimension 4. This result also holds for Lorentzian-Einstein metrics with a positive cosmological
constant.
1. Introduction.
Let M be the interior of a compact (n + 1)-dimensional manifold ŻM with non-empty boundary
M. A complete metric g on M is Cm, conformally compact if there is a defining function on
ŻM such that the conformally equivalent metric
(1.1) g = 2
g
extends to a Cm, Riemannian metric on the compactification ŻM. A defining function is a smooth,
non-negative function on ŻM with -1(0) = M and d = 0 on M.
The induced Riemannian metric = g|M is called the boundary metric associated to the
compactification g. Since there are many possible defining functions, and hence many conformal
compactifications of a given metric g, only the conformal class [] of on M is uniquely deter-

  

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

 

Collections: Mathematics