Commun. Math. Phys. 162, 371-397 (1994)
Analysis of the static spherically symmetric
SU(n)-Einstein-Yang-Mills equations

H.P. Künzle

Department of Mathematics, University of Alberta
Edmonton, Canada T6G~2G1
e-mail:HP.Kunzle@UAlberta.ca

Abstract

The singular boundary value problem that arises for the static spherically symmetric SU(n)-Einstein-Yang-Mills equations in the so-called magnetic case is analyzed. Among the possible actions of SU(2) on a SU(n)-principal bundles over space-time there is one which appears to be the most natural. If one assumes that no electrostatic type component is present in the Yang-Mills fields and the gauge is suitably fixed a set of n-1 second order and two first order differential equations is obtained for n-1 gauge potentials and two metric components as functions of the radial distance. This system generalizes the one for the case n=2 that lead to the discrete series of the Bartnik-Mckinnon and the corresponding black hole solutions. It is highly nonlinear and singular at r=infinity and at r=0 or at the black hole horizon but it is known to admit at least one series of discrete solutions which are scaled versions of the n=2 case. In this paper local existence and uniqueness of solutions near these singular points is established which turns out to be a nontrivial problem for general n. Moreover, a number of new numerical soliton (i.e. globally regular) numerical solutions of the SU(3)-EYM equations are found that are not scaled n=2 solutions.