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EXTENSION OF SYMMETRIES ON EINSTEIN MANIFOLDS WITH BOUNDARY
 

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 Mn+1 be a compact (n + 1)-dimensional manifold-with-boundary, and suppose g is a (Rie-
mannian) Einstein metric on M, so that
(1.1) Ricg = 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 = S3 \ (B1 B2), where Bi are a pair of
disjoint round 3-balls in S3 endowed with a round metric; then a generic pair of Killing fields Xi
on S2
i = Bi does not extend to a Killing field on M. Also, setting M = T3 \B where B is a round

  

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

 

Collections: Mathematics