Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field over S⁴ with group G=SU₂, SU₃ or U₂ is either self-dual or anti-self-dual. The analogous statement for SO₄-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicit C⁰-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the group G does not matter, neither is the dimension of the base manifold constrained to be four.