Small coupling limit and multiple solutions to the Dirichlet Problem for Yang-Mills connections in 4 dimensions - Part III

In this paper, the third of this series, we prove that the spaces A,p k (A0; q) and B,p 0,k(A0; q) which contain Lp k-approximate solutions to the Dirichlet problem for the -Yang Mills equations on a four dimensional disk B4, carry a natural manifold structure (more precisely a natural structure of Banach bundle), for p(k+1) > 4. All results apply also if B4 is replaced by a general compact manifold with boundary, and SU(2) is replaced by any compact Lie group. We also construct bases for the tangent space to the space of approximate solutions, thus showing that this space is 8-dimensional for sufficiently small, and prove some technical results used in Parts I and II for the proof of the existence of multiple solution and, in particular, non-minimal ones, for this non-compact variational problem.