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 *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.