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 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  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  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 . 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 . 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
Alan D. Rendall
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