Summary: EXTENSION OF SYMMETRIES ON EINSTEIN MANIFOLDS
WITH BOUNDARY
MICHAEL T. ANDERSON
Abstract. We investigate the validity of the isometry extension property for (Riemannian) Ein­
stein metrics on compact manifolds M with boundary #M . Given a metric # on #M , this is the
issue of whether any Killing field X of (#M, #) extends to a Killing field of any Einstein metric
(M, g) bounding (#M, #). Under a mild condition on the fundamental group, this is proved to be
the case at least when X preserves the mean curvature of #M in (M, g).
1. Introduction.
Let M n+1 be a compact (n + 1)­dimensional manifold­with­boundary, and suppose g is a (Rie­
mannian) Einstein metric on M , so that
(1.1) Ric g = #g,
for some constant # # R. The metric g induces a Riemannian boundary metric # on #M . In this
paper we consider the issue of whether isometries of the boundary structure (#M, #) necessarily
extend to isometries of any filling Einstein manifold (M, g).
In general, without any assumptions, this isometry extension property will not hold. It is false
for instance if #M is not connected. For example, let M = S 3
\ (B 1 # B 2 ), where B i are a pair of
disjoint round 3­balls in S 3 endowed with a round metric; then a generic pair of Killing fields X i
on S 2

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

Collections: Mathematics