Summary: ON THE STRUCTURE OF CONFORMALLY COMPACT EINSTEIN METRICS
MICHAEL T. ANDERSON
Abstract. Let M be an (n+1)­dimensional manifold with non­empty 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 non­empty. 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 stress­energy tensor at conformal infinity, again in
dimension 4. This result also holds for Lorentzian­Einstein metrics with a positive cosmological
constant.
1. Introduction.
Let M be the interior of a compact (n + 1)­dimensional manifold •
M with non­empty boundary
#M . A complete metric g on M is C m,# conformally compact if there is a defining function # on
•
M such that the conformally equivalent metric
(1.1)
# g = # 2 g
extends to a C m,# Riemannian metric on the compactification •
M . A defining function # is a smooth,
non­negative function on •

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

Collections: Mathematics