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Analysis of the static spherically symmetricSU(n)-Einstein-Yang-Mills equations

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Abstract

The singular boundary value problem that arises for the static spherically symmetricSU(n)-Einstein-Yang-Mills equations in the so-called magnetic case is analyzed. Among the possible actions ofSU(2) on aSU(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 ofn-1 second order and two first order differential equations is obtained forn-1 gauge potentials and two metric components as functions of the radial distance. This system generalizes the one for the casen=2 that leads to the discrete series of the Bartnick-Mckinnon and the corresponding black hole solutions. It is highly nonlinear and singular atr=∞ and atr=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 then=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 generaln. Moreover, a number of new numerical soliton (i.e. globally regular) numerical solutions of theSU(3)-EYM equations are found that are not scaledn=2 solutions.

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Authors and Affiliations

  1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Canada

    H. P. Künzle

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  1. H. P. Künzle
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Communicated by S.-T. Yau

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Künzle, H.P. Analysis of the static spherically symmetricSU(n)-Einstein-Yang-Mills equations. Commun.Math. Phys. 162, 371–397 (1994). https://doi.org/10.1007/BF02102023

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  • DOI: https://doi.org/10.1007/BF02102023

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