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UNIQUE CONTINUATION RESULTS FOR RICCI CURVATURE AND APPLICATIONS
 

Summary: UNIQUE CONTINUATION RESULTS FOR RICCI CURVATURE AND
APPLICATIONS
MICHAEL T. ANDERSON AND MARC HERZLICH
Abstract. Unique continuation results are proved for metrics with prescribed Ricci curvature in
the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete
conformally compact metrics on such manifolds. Related to this issue, an isometry extension
property is proved: continuous groups of isometries at conformal in nity extend into the bulk of
any complete conformally compact Einstein metric. Relations of this property with the invariance
of the Gauss-Codazzi constraint equations under deformations are also discussed.
1. Introduction.
In this paper, we study certain issues related to the boundary behavior of metrics with prescribed
Ricci curvature. Let M be a compact (n + 1)-dimensional manifold with compact non-empty
boundary @M . We consider two possible classes of Riemannian metrics g on M . First, g may
extend smoothly to a Riemannian metric on the closure 
M = M [@M , thus inducing a Riemannian
metric = gj @M on @M . Second, g may be a complete metric on M , so that @M is \at in nity".
In this case, we assume that g is conformally compact, i.e. there exists a de ning function  for
@M in M such that the conformally equivalent metric
(1.1) eg =  2 g
extends at least C 2 to @M . The de ning function  is unique only up to multiplication by posi-

  

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

 

Collections: Mathematics