The Yang-Mills equations were introduced by theoretical physicists and are now accepted as a basic ingredient in particle theory. In the past decade these equations have become important in mathematics in two separate areas. It was observed early on that the twistor formalism of Penrose and the Atiyah-Singer index theory can be employed to good purpose in order to describe special The theory of holomorphic vector bundles is central to algebraic geometry. The concept of a stable vector bundle was introduced by Mumford in h s study of moduli for bundles [18]. This theme has been pursued by several mathematicians, notably Takemoto, Horrocks, Gieseker, Maruyama and Barth. An important contribution was made by Bogomolov [4], who showed that, for a projective variety M with Pic ( M ) = Z and E a stable bundle on M , the inequality k -1is valid. Here k = rank E , c, and c2 are the first and second Chern classes, and w is a polarization for which E is stable.Bogomolov's work inspired the work of Miyaoka [17] on the inequality 3c, 2 c: for algebraic surfaces of general type. Independently, the second author also gave a proof of this inequality at the same time. While the methods of Miyaoka and Bogomolov are essentially based on algebraic geometry, the method of the second author used the techniques of partial differential equations to

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Table of ContentsPART 1 2 1. Basic formulas and characteristic notation 2 2. The canonical associated linear problem or Lax pair 5 3. The Dolan representation 8 4. The variation formulas for the extended solution 12 5. The representation of $/ (S 2 , G) on holomorphic maps C* -• G 14 6. The representation of J#k(S 2 , G) on extended harmonic maps and Backlund transformations 20 7. The additional S ι action 23 8. Harmonic maps into Grassmannians 24 PART II 27 9. The single uniton 27 10. The fixed points of the S ι action 31 11. Global conservation laws and finiteness 35 12. Adding a uniton by singular Backlund transformation 37 13. The minimal uniton number 40 14. The unique factorization theorem 42 15. Complex Grassmannian manifolds again 45 16. Additional questions and problems 47

Abstract.We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball in R" when the integral L "/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds with L p integral norms bounded, p > n/2.

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