The area of stationary solutions of the Einstein equations
coupled to field theoretic matter models has been active in
recent years as a consequence of the discovery by Bartnik and
McKinnon[5] of a discrete family of regular static spherically symmetric
solutions of the Einstein-Yang-Mills equations with gauge group
*SU*
(2). The equations to be solved are ordinary differential
equations and in [5] they were solved numerically by a shooting method. The first
existence proof for a solution of this kind is due to Smoller,
Wasserman, Yau and McLeod [75] and involves an arduous qualitative analysis of the
differential equations. The work on the Bartnik-McKinnon
solutions, including the existence theorems, has been extended in
many directions. Recently static solutions of the
Einstein-Yang-Mills equations which are not spherically symmetric
were discovered numerically [56]. It is a challenge to prove the existence of solutions of this
kind. Now the ordinary differential equations of the previously
known case are replaced by elliptic equations. Moreover, the
solutions appear to still be discrete, so that a simple
perturbation argument starting from the spherical case does not
seem feasible. In another development it was shown that a
linearized analysis indicates the existence of stationary
non-static solutions [10]. It would be desirable to study the question of linearization
stability in this case, which, if the answer were favourable,
would give an existence proof for solutions of this kind.

Local and global existence theorems for the Einstein
equations
Alan D. Rendall
http://www.livingreviews.org/lrr-1998-4
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