Regularity theory for the generalized Neumann problem for Yang-Mills Connections - Non-trivial examples in dimensions 3 and 4

We develop the existence acid regularity theory fur the generalized Neumann problem for Yang-Mills: connections. This is the most general boundary val ue problem for connections on a compact manifold with smooth boundary, with geometric meaning. It is obtained by reflecting the base manifold across i ts boundry and lifting this action non-trivially to the bundle. The proscri bed lifting corresponds to a geometric invariant, which is similar to the m onopole number. When this invariant is non-zero, there exist non-trivial so lutions of the generalized Neumann problem. We Drove the existence of non-t rivial solutions over the 3-dimensional disk D-3 and over the 4-dimensional manifold D-3 x S-1. We outline the procedure for finding non-trivial examp les of solutions over more general manifolds: of dimension 3 and 4.