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 its series, we prove that the sobolev spaces of L^p_k approximate solutions to the Dirichlet problem for the epsilon-Yang Mills equations on a four dimensional disk, carry a natural manifold structure (more precisely a natural structure of Banach bundle), for p(k+1)> 4. All results apply also if the four-dimensional disk 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 epsilon 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.