2. — General solutions of the Yang-Mills equations.

In these lectures I have concentrated on the instanton problems which correspond to absolute minima of the Yang-Mills functional on S4. I shall now review what is known about general solutions of the Yang-Mills equa­tions, corresponding to critical points which are not absolute minima.

First of all there is a recent result of Bourguignon, Lawson and Simons [11] which shows that there are no other local minima. It is at present unknown if other critical points exist, but [11] asserts that any such points must be unstable. The proof consists in showing that the second variation is indefinite.

The twistor interpretation of instantons does not immediately apply to other Yang-Mills solutions, but Witten [44] and Green et al. [24] have shown how to generalize Ward's ideas to the general case. However, the twistor interpretation is now more complicated and its potential has not yet been exploited.

Topological aspects of Yang-Mills theory related to ideas of Morse theory have been studied in [5]. In this and other respects there are close analogies with the non-linear a model in 2 dimensions and these analogies suggest that no other critical points exist.

Finally, K. Uhlenbeck has recently shown [41] that square integrability on Ti* for Yang-Mills solutions automatically ensures the extension to 8*. More precisely there is a purely local result asserting that a smooth con­nection defined in a neighbourhood of 0 (but not at 0), which is locally square integrable and satisfies the Yang-Mills equations, automatically extends to a smooth connection defined at 0. In other words, point sin­gularities are « removable ». Applied to the point at oo, after a conformal transformation, this yields the extension from Rl to 81.